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Calculated Shard: 63 (from laksa180)

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curl -X POST \
  'http://laksa063.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.emathhelp.net/calculators/linear-algebra/eigenvalue-and-eigenvector-calculator/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.emathhelp.net/calculators/linear-algebra/eigenvalue-and-eigenvector-calculator/\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.emathhelp.net\/calculators\/linear-algebra\/eigenvalue-and-eigenvector-calculator\/","crawl_time":1775469914,"first_indexed_time":1487764359,"http_code":200,"src_unparsed":"net,emathhelp!www,\/calculators\/linear-algebra\/eigenvalue-and-eigenvector-calculator\/ s443","src_root_hash":"16951690241190635063","history_drop_reason":null,"meta_title":"Eigenvalues and Eigenvectors Calculator - eMathHelp","meta_descriptions":["The calculator will find the eigenvalues and eigenvectors (eigenspace) of the given square matrix, with steps shown."],"attrs_boilerpipe_text":"The calculator will find the eigenvalues and eigenvectors (eigenspace) of the given square matrix, with steps shown.\nRelated calculator:\nCharacteristic Polynomial Calculator\nYour Input\nFind the eigenvalues and eigenvectors of\n$$$\n\\left[\\begin{array}{cc}1 & 2\\\\0 & 3\\end{array}\\right]\n$$$\n.\nSolution\nStart from forming a new matrix by subtracting\n$$$\n\\lambda\n$$$\nfrom the diagonal entries of the given matrix:\n$$$\n\\left[\\begin{array}{cc}1 - \\lambda & 2\\\\0 & 3 - \\lambda\\end{array}\\right]\n$$$\n.\nThe determinant of the obtained matrix is\n$$$\n\\left(\\lambda - 3\\right) \\left(\\lambda - 1\\right)\n$$$\n(for steps, see\ndeterminant calculator\n).\nSolve the equation\n$$$\n\\left(\\lambda - 3\\right) \\left(\\lambda - 1\\right) = 0\n$$$\n.\nThe roots are\n$$$\n\\lambda_{1} = 3\n$$$\n,\n$$$\n\\lambda_{2} = 1\n$$$\n(for steps, see\nequation solver\n).\nThese are the eigenvalues.\nNext, find the eigenvectors.\n$$$\n\\lambda = 3\n$$$\n$$$\n\\left[\\begin{array}{cc}1 - \\lambda & 2\\\\0 & 3 - \\lambda\\end{array}\\right] = \\left[\\begin{array}{cc}-2 & 2\\\\0 & 0\\end{array}\\right]\n$$$\nThe null space of this matrix is\n$$$\n\\left\\{\\left[\\begin{array}{c}1\\\\1\\end{array}\\right]\\right\\}\n$$$\n(for steps, see\nnull space calculator\n).\nThis is the eigenvector.\n$$$\n\\lambda = 1\n$$$\n$$$\n\\left[\\begin{array}{cc}1 - \\lambda & 2\\\\0 & 3 - \\lambda\\end{array}\\right] = \\left[\\begin{array}{cc}0 & 2\\\\0 & 2\\end{array}\\right]\n$$$\nThe null space of this matrix is\n$$$\n\\left\\{\\left[\\begin{array}{c}1\\\\0\\end{array}\\right]\\right\\}\n$$$\n(for steps, see\nnull space calculator\n).\nThis is the eigenvector.\nAnswer\nEigenvalue:\n$$$\n3\n$$$\nA\n, multiplicity:\n$$$\n1\n$$$\nA\n, eigenvector:\n$$$\n\\left[\\begin{array}{c}1\\\\1\\end{array}\\right]\n$$$\nA\n.\nEigenvalue:\n$$$\n1\n$$$\nA\n, multiplicity:\n$$$\n1\n$$$\nA\n, eigenvector:\n$$$\n\\left[\\begin{array}{c}1\\\\0\\end{array}\\right]\n$$$\nA\n.","attrs_markdown":"eMathHelp works best with JavaScript enabled\n\n[![eMathHelp](https:\/\/www.emathhelp.net\/static\/assets\/images\/logo.c8153e69673e.png)](https:\/\/www.emathhelp.net\/)\n\n![No AI is used](https:\/\/www.emathhelp.net\/static\/no_ai.367f5673513f.svg)\n\n- [Math Calculator](https:\/\/www.emathhelp.net\/math-calculator\/)\n- [Calculators](https:\/\/www.emathhelp.net\/calculators\/)\n- [Notes](https:\/\/www.emathhelp.net\/notes\/)\n- [Games](https:\/\/www.emathhelp.net\/games-and-logic-puzzles\/)\n\n- [Algebra](https:\/\/www.emathhelp.net\/algebra-calculator\/)\n- [Geometry](https:\/\/www.emathhelp.net\/geometry-calculator\/)\n- [Pre-Calculus](https:\/\/www.emathhelp.net\/pre-calculus-calculator\/)\n- [Calculus](https:\/\/www.emathhelp.net\/calculus-calculator\/)\n- [Linear Algebra](https:\/\/www.emathhelp.net\/linear-algebra-calculator\/)\n- [Discrete Math](https:\/\/www.emathhelp.net\/discrete-mathematics-calculator\/)\n- [Probability\/Statistics](https:\/\/www.emathhelp.net\/probability-statistics-calculator\/)\n- [Linear Programming](https:\/\/www.emathhelp.net\/linear-programming-calculator\/)\n\n1. [Home](https:\/\/www.emathhelp.net\/)\n2. [Calculators](https:\/\/www.emathhelp.net\/calculators\/)\n3. [Calculators: Linear Algebra](https:\/\/www.emathhelp.net\/calculators\/linear-algebra\/)\n4. [Linear Algebra Calculator](https:\/\/www.emathhelp.net\/linear-algebra-calculator\/)\n# Eigenvalues and Eigenvectors Calculator\n## Calculate eigenvalues and eigenvectors step by step\nThe calculator will find the eigenvalues and eigenvectors (eigenspace) of the given square matrix, with steps shown.\n\nRelated calculator: [Characteristic Polynomial Calculator](https:\/\/www.emathhelp.net\/calculators\/linear-algebra\/characteristic-polynomial-calculator\/)\n\n### Your Input\n**Find the eigenvalues and eigenvectors of \\$\\$\\$\\\\left\\[\\\\begin{array}{cc}1 & 2\\\\\\\\0 & 3\\\\end{array}\\\\right\\]\\$\\$\\$.**\n\n### Solution\nStart from forming a new matrix by subtracting \\$\\$\\$\\\\lambda\\$\\$\\$ from the diagonal entries of the given matrix: \\$\\$\\$\\\\left\\[\\\\begin{array}{cc}1 - \\\\lambda & 2\\\\\\\\0 & 3 - \\\\lambda\\\\end{array}\\\\right\\]\\$\\$\\$.\n\nThe determinant of the obtained matrix is \\$\\$\\$\\\\left(\\\\lambda - 3\\\\right) \\\\left(\\\\lambda - 1\\\\right)\\$\\$\\$ (for steps, see [determinant calculator](https:\/\/www.emathhelp.net\/calculators\/linear-algebra\/matrix-determinant-calculator\/?i=%5B%5B1+-+lambda%2C2%5D%2C%5B0%2C3+-+lambda%5D%5D \"determinant calculator\")).\n\nSolve the equation \\$\\$\\$\\\\left(\\\\lambda - 3\\\\right) \\\\left(\\\\lambda - 1\\\\right) = 0\\$\\$\\$.\n\nThe roots are \\$\\$\\$\\\\lambda\\_{1} = 3\\$\\$\\$, \\$\\$\\$\\\\lambda\\_{2} = 1\\$\\$\\$ (for steps, see [equation solver](https:\/\/www.emathhelp.net\/calculators\/algebra-2\/equation-solver-calculator\/?f=%28lambda+-+3%29%2A%28lambda+-+1%29&var=lambda \"equation solver\")).\n\nThese are the eigenvalues.\n\nNext, find the eigenvectors.\n\n- \\$\\$\\$\\\\lambda = 3\\$\\$\\$\n  \\$\\$\\$\\\\left\\[\\\\begin{array}{cc}1 - \\\\lambda & 2\\\\\\\\0 & 3 - \\\\lambda\\\\end{array}\\\\right\\] = \\\\left\\[\\\\begin{array}{cc}-2 & 2\\\\\\\\0 & 0\\\\end{array}\\\\right\\]\\$\\$\\$\n  The null space of this matrix is \\$\\$\\$\\\\left\\\\{\\\\left\\[\\\\begin{array}{c}1\\\\\\\\1\\\\end{array}\\\\right\\]\\\\right\\\\}\\$\\$\\$ (for steps, see [null space calculator](https:\/\/www.emathhelp.net\/calculators\/linear-algebra\/null-space-calculator\/?i=%5B%5B-2%2C2%5D%2C%5B0%2C0%5D%5D \"null space calculator\")).\n  This is the eigenvector.\n- \\$\\$\\$\\\\lambda = 1\\$\\$\\$\n  \\$\\$\\$\\\\left\\[\\\\begin{array}{cc}1 - \\\\lambda & 2\\\\\\\\0 & 3 - \\\\lambda\\\\end{array}\\\\right\\] = \\\\left\\[\\\\begin{array}{cc}0 & 2\\\\\\\\0 & 2\\\\end{array}\\\\right\\]\\$\\$\\$\n  The null space of this matrix is \\$\\$\\$\\\\left\\\\{\\\\left\\[\\\\begin{array}{c}1\\\\\\\\0\\\\end{array}\\\\right\\]\\\\right\\\\}\\$\\$\\$ (for steps, see [null space calculator](https:\/\/www.emathhelp.net\/calculators\/linear-algebra\/null-space-calculator\/?i=%5B%5B0%2C2%5D%2C%5B0%2C2%5D%5D \"null space calculator\")).\n  This is the eigenvector.\n\n### Answer\n**Eigenvalue: \\$\\$\\$3\\$\\$\\$A, multiplicity: \\$\\$\\$1\\$\\$\\$A, eigenvector: \\$\\$\\$\\\\left\\[\\\\begin{array}{c}1\\\\\\\\1\\\\end{array}\\\\right\\]\\$\\$\\$A.**\n\n**Eigenvalue: \\$\\$\\$1\\$\\$\\$A, multiplicity: \\$\\$\\$1\\$\\$\\$A, eigenvector: \\$\\$\\$\\\\left\\[\\\\begin{array}{c}1\\\\\\\\0\\\\end{array}\\\\right\\]\\$\\$\\$A.**\n\n- [About](https:\/\/www.emathhelp.net\/about\/)\n- [Contact](https:\/\/www.emathhelp.net\/contact\/)\n- [Terms of use](https:\/\/www.emathhelp.net\/terms-of-use\/)\n- [Privacy Policy](https:\/\/www.emathhelp.net\/privacy-policy\/)\n\nCopyright (c) 2026. All rights reserved.","attrs_readable_markdown":"The calculator will find the eigenvalues and eigenvectors (eigenspace) of the given square matrix, with steps shown.\n\nRelated calculator: [Characteristic Polynomial Calculator](https:\/\/www.emathhelp.net\/calculators\/linear-algebra\/characteristic-polynomial-calculator\/)\n\nYour Input\n\n**Find the eigenvalues and eigenvectors of \\$\\$\\$\\\\left\\[\\\\begin{array}{cc}1 & 2\\\\\\\\0 & 3\\\\end{array}\\\\right\\]\\$\\$\\$.**\n\nSolution\n\nStart from forming a new matrix by subtracting \\$\\$\\$\\\\lambda\\$\\$\\$ from the diagonal entries of the given matrix: \\$\\$\\$\\\\left\\[\\\\begin{array}{cc}1 - \\\\lambda & 2\\\\\\\\0 & 3 - \\\\lambda\\\\end{array}\\\\right\\]\\$\\$\\$.\n\nThe determinant of the obtained matrix is \\$\\$\\$\\\\left(\\\\lambda - 3\\\\right) \\\\left(\\\\lambda - 1\\\\right)\\$\\$\\$ (for steps, see [determinant calculator](https:\/\/www.emathhelp.net\/calculators\/linear-algebra\/matrix-determinant-calculator\/?i=%5B%5B1+-+lambda%2C2%5D%2C%5B0%2C3+-+lambda%5D%5D \"determinant calculator\")).\n\nSolve the equation \\$\\$\\$\\\\left(\\\\lambda - 3\\\\right) \\\\left(\\\\lambda - 1\\\\right) = 0\\$\\$\\$.\n\nThe roots are \\$\\$\\$\\\\lambda\\_{1} = 3\\$\\$\\$, \\$\\$\\$\\\\lambda\\_{2} = 1\\$\\$\\$ (for steps, see [equation solver](https:\/\/www.emathhelp.net\/calculators\/algebra-2\/equation-solver-calculator\/?f=%28lambda+-+3%29%2A%28lambda+-+1%29&var=lambda \"equation solver\")).\n\nThese are the eigenvalues.\n\nNext, find the eigenvectors.\n\n- \\$\\$\\$\\\\lambda = 3\\$\\$\\$\n  \\$\\$\\$\\\\left\\[\\\\begin{array}{cc}1 - \\\\lambda & 2\\\\\\\\0 & 3 - \\\\lambda\\\\end{array}\\\\right\\] = \\\\left\\[\\\\begin{array}{cc}-2 & 2\\\\\\\\0 & 0\\\\end{array}\\\\right\\]\\$\\$\\$\n  The null space of this matrix is \\$\\$\\$\\\\left\\\\{\\\\left\\[\\\\begin{array}{c}1\\\\\\\\1\\\\end{array}\\\\right\\]\\\\right\\\\}\\$\\$\\$ (for steps, see [null space calculator](https:\/\/www.emathhelp.net\/calculators\/linear-algebra\/null-space-calculator\/?i=%5B%5B-2%2C2%5D%2C%5B0%2C0%5D%5D \"null space calculator\")).\n  This is the eigenvector.\n- \\$\\$\\$\\\\lambda = 1\\$\\$\\$\n  \\$\\$\\$\\\\left\\[\\\\begin{array}{cc}1 - \\\\lambda & 2\\\\\\\\0 & 3 - \\\\lambda\\\\end{array}\\\\right\\] = \\\\left\\[\\\\begin{array}{cc}0 & 2\\\\\\\\0 & 2\\\\end{array}\\\\right\\]\\$\\$\\$\n  The null space of this matrix is \\$\\$\\$\\\\left\\\\{\\\\left\\[\\\\begin{array}{c}1\\\\\\\\0\\\\end{array}\\\\right\\]\\\\right\\\\}\\$\\$\\$ (for steps, see [null space calculator](https:\/\/www.emathhelp.net\/calculators\/linear-algebra\/null-space-calculator\/?i=%5B%5B0%2C2%5D%2C%5B0%2C2%5D%5D \"null space calculator\")).\n  This is the eigenvector.\n\nAnswer\n\n**Eigenvalue: \\$\\$\\$3\\$\\$\\$A, multiplicity: \\$\\$\\$1\\$\\$\\$A, eigenvector: \\$\\$\\$\\\\left\\[\\\\begin{array}{c}1\\\\\\\\1\\\\end{array}\\\\right\\]\\$\\$\\$A.**\n\n**Eigenvalue: \\$\\$\\$1\\$\\$\\$A, multiplicity: \\$\\$\\$1\\$\\$\\$A, eigenvector: \\$\\$\\$\\\\left\\[\\\\begin{array}{c}1\\\\\\\\0\\\\end{array}\\\\right\\]\\$\\$\\$A.**","meta_canonical":null}

3. Robots.txt Check

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4. Spam/Ban Check

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5. Seen Status Check

ℹ️ Skipped - page is already crawled

📄

INDEXABLE

✅

CRAWLED

9 hours ago

🤖

ROBOTS ALLOWED

Page Info Filters

Filter	Status	Condition	Details
HTTP status	PASS	`download_http_code = 200`	HTTP 200
Age cutoff	PASS	`download_stamp > now() - 6 MONTH`	0 months ago
History drop	PASS	`isNull(history_drop_reason)`	No drop reason
Spam/ban	PASS	`fh_dont_index != 1 AND ml_spam_score = 0`	ml_spam_score=0
Canonical	PASS	`meta_canonical IS NULL OR = '' OR = src_unparsed`	Not set

Page Details

Property	Value
URL	https://www.emathhelp.net/calculators/linear-algebra/eigenvalue-and-eigenvector-calculator/
Last Crawled	2026-04-06 10:05:14 (9 hours ago)
First Indexed	2017-02-22 11:52:39 (9 years ago)
HTTP Status Code	200
Meta Title	Eigenvalues and Eigenvectors Calculator - eMathHelp
Meta Description	The calculator will find the eigenvalues and eigenvectors (eigenspace) of the given square matrix, with steps shown.
Meta Canonical	null
Boilerpipe Text	The calculator will find the eigenvalues and eigenvectors (eigenspace) of the given square matrix, with steps shown. Related calculator: Characteristic Polynomial Calculator Your Input Find the eigenvalues and eigenvectors of $$$ \left[\begin{array}{cc}1 & 2\\0 & 3\end{array}\right] $$$ . Solution Start from forming a new matrix by subtracting $$$ \lambda $$$ from the diagonal entries of the given matrix: $$$ \left[\begin{array}{cc}1 - \lambda & 2\\0 & 3 - \lambda\end{array}\right] $$$ . The determinant of the obtained matrix is $$$ \left(\lambda - 3\right) \left(\lambda - 1\right) $$$ (for steps, see determinant calculator ). Solve the equation $$$ \left(\lambda - 3\right) \left(\lambda - 1\right) = 0 $$$ . The roots are $$$ \lambda_{1} = 3 $$$ , $$$ \lambda_{2} = 1 $$$ (for steps, see equation solver ). These are the eigenvalues. Next, find the eigenvectors. $$$ \lambda = 3 $$$ $$$ \left[\begin{array}{cc}1 - \lambda & 2\\0 & 3 - \lambda\end{array}\right] = \left[\begin{array}{cc}-2 & 2\\0 & 0\end{array}\right] $$$ The null space of this matrix is $$$ \left\{\left[\begin{array}{c}1\\1\end{array}\right]\right\} $$$ (for steps, see null space calculator ). This is the eigenvector. $$$ \lambda = 1 $$$ $$$ \left[\begin{array}{cc}1 - \lambda & 2\\0 & 3 - \lambda\end{array}\right] = \left[\begin{array}{cc}0 & 2\\0 & 2\end{array}\right] $$$ The null space of this matrix is $$$ \left\{\left[\begin{array}{c}1\\0\end{array}\right]\right\} $$$ (for steps, see null space calculator ). This is the eigenvector. Answer Eigenvalue: $$$ 3 $$$ A , multiplicity: $$$ 1 $$$ A , eigenvector: $$$ \left[\begin{array}{c}1\\1\end{array}\right] $$$ A . Eigenvalue: $$$ 1 $$$ A , multiplicity: $$$ 1 $$$ A , eigenvector: $$$ \left[\begin{array}{c}1\\0\end{array}\right] $$$ A .
Markdown	eMathHelp works best with JavaScript enabled [![eMathHelp](https://www.emathhelp.net/static/assets/images/logo.c8153e69673e.png)](https://www.emathhelp.net/) ![No AI is used](https://www.emathhelp.net/static/no_ai.367f5673513f.svg) - [Math Calculator](https://www.emathhelp.net/math-calculator/) - [Calculators](https://www.emathhelp.net/calculators/) - [Notes](https://www.emathhelp.net/notes/) - [Games](https://www.emathhelp.net/games-and-logic-puzzles/) - [Algebra](https://www.emathhelp.net/algebra-calculator/) - [Geometry](https://www.emathhelp.net/geometry-calculator/) - [Pre-Calculus](https://www.emathhelp.net/pre-calculus-calculator/) - [Calculus](https://www.emathhelp.net/calculus-calculator/) - [Linear Algebra](https://www.emathhelp.net/linear-algebra-calculator/) - [Discrete Math](https://www.emathhelp.net/discrete-mathematics-calculator/) - [Probability/Statistics](https://www.emathhelp.net/probability-statistics-calculator/) - [Linear Programming](https://www.emathhelp.net/linear-programming-calculator/) 1. [Home](https://www.emathhelp.net/) 2. [Calculators](https://www.emathhelp.net/calculators/) 3. [Calculators: Linear Algebra](https://www.emathhelp.net/calculators/linear-algebra/) 4. [Linear Algebra Calculator](https://www.emathhelp.net/linear-algebra-calculator/) # Eigenvalues and Eigenvectors Calculator ## Calculate eigenvalues and eigenvectors step by step The calculator will find the eigenvalues and eigenvectors (eigenspace) of the given square matrix, with steps shown. Related calculator: [Characteristic Polynomial Calculator](https://www.emathhelp.net/calculators/linear-algebra/characteristic-polynomial-calculator/) ### Your Input Find the eigenvalues and eigenvectors of \$\$\$\\left\[\\begin{array}{cc}1 & 2\\\\0 & 3\\end{array}\\right\]\$\$\$. ### Solution Start from forming a new matrix by subtracting \$\$\$\\lambda\$\$\$ from the diagonal entries of the given matrix: \$\$\$\\left\[\\begin{array}{cc}1 - \\lambda & 2\\\\0 & 3 - \\lambda\\end{array}\\right\]\$\$\$. The determinant of the obtained matrix is \$\$\$\\left(\\lambda - 3\\right) \\left(\\lambda - 1\\right)\$\$\$ (for steps, see [determinant calculator](https://www.emathhelp.net/calculators/linear-algebra/matrix-determinant-calculator/?i=%5B%5B1+-+lambda%2C2%5D%2C%5B0%2C3+-+lambda%5D%5D "determinant calculator")). Solve the equation \$\$\$\\left(\\lambda - 3\\right) \\left(\\lambda - 1\\right) = 0\$\$\$. The roots are \$\$\$\\lambda\_{1} = 3\$\$\$, \$\$\$\\lambda\_{2} = 1\$\$\$ (for steps, see [equation solver](https://www.emathhelp.net/calculators/algebra-2/equation-solver-calculator/?f=%28lambda+-+3%29%2A%28lambda+-+1%29&var=lambda "equation solver")). These are the eigenvalues. Next, find the eigenvectors. - \$\$\$\\lambda = 3\$\$\$ \$\$\$\\left\[\\begin{array}{cc}1 - \\lambda & 2\\\\0 & 3 - \\lambda\\end{array}\\right\] = \\left\[\\begin{array}{cc}-2 & 2\\\\0 & 0\\end{array}\\right\]\$\$\$ The null space of this matrix is \$\$\$\\left\\{\\left\[\\begin{array}{c}1\\\\1\\end{array}\\right\]\\right\\}\$\$\$ (for steps, see [null space calculator](https://www.emathhelp.net/calculators/linear-algebra/null-space-calculator/?i=%5B%5B-2%2C2%5D%2C%5B0%2C0%5D%5D "null space calculator")). This is the eigenvector. - \$\$\$\\lambda = 1\$\$\$ \$\$\$\\left\[\\begin{array}{cc}1 - \\lambda & 2\\\\0 & 3 - \\lambda\\end{array}\\right\] = \\left\[\\begin{array}{cc}0 & 2\\\\0 & 2\\end{array}\\right\]\$\$\$ The null space of this matrix is \$\$\$\\left\\{\\left\[\\begin{array}{c}1\\\\0\\end{array}\\right\]\\right\\}\$\$\$ (for steps, see [null space calculator](https://www.emathhelp.net/calculators/linear-algebra/null-space-calculator/?i=%5B%5B0%2C2%5D%2C%5B0%2C2%5D%5D "null space calculator")). This is the eigenvector. ### Answer Eigenvalue: \$\$\$3\$\$\$A, multiplicity: \$\$\$1\$\$\$A, eigenvector: \$\$\$\\left\[\\begin{array}{c}1\\\\1\\end{array}\\right\]\$\$\$A. Eigenvalue: \$\$\$1\$\$\$A, multiplicity: \$\$\$1\$\$\$A, eigenvector: \$\$\$\\left\[\\begin{array}{c}1\\\\0\\end{array}\\right\]\$\$\$A. - [About](https://www.emathhelp.net/about/) - [Contact](https://www.emathhelp.net/contact/) - [Terms of use](https://www.emathhelp.net/terms-of-use/) - [Privacy Policy](https://www.emathhelp.net/privacy-policy/) Copyright (c) 2026. All rights reserved.
Readable Markdown	The calculator will find the eigenvalues and eigenvectors (eigenspace) of the given square matrix, with steps shown. Related calculator: [Characteristic Polynomial Calculator](https://www.emathhelp.net/calculators/linear-algebra/characteristic-polynomial-calculator/) Your Input Find the eigenvalues and eigenvectors of \$\$\$\\left\[\\begin{array}{cc}1 & 2\\\\0 & 3\\end{array}\\right\]\$\$\$. Solution Start from forming a new matrix by subtracting \$\$\$\\lambda\$\$\$ from the diagonal entries of the given matrix: \$\$\$\\left\[\\begin{array}{cc}1 - \\lambda & 2\\\\0 & 3 - \\lambda\\end{array}\\right\]\$\$\$. The determinant of the obtained matrix is \$\$\$\\left(\\lambda - 3\\right) \\left(\\lambda - 1\\right)\$\$\$ (for steps, see [determinant calculator](https://www.emathhelp.net/calculators/linear-algebra/matrix-determinant-calculator/?i=%5B%5B1+-+lambda%2C2%5D%2C%5B0%2C3+-+lambda%5D%5D "determinant calculator")). Solve the equation \$\$\$\\left(\\lambda - 3\\right) \\left(\\lambda - 1\\right) = 0\$\$\$. The roots are \$\$\$\\lambda\_{1} = 3\$\$\$, \$\$\$\\lambda\_{2} = 1\$\$\$ (for steps, see [equation solver](https://www.emathhelp.net/calculators/algebra-2/equation-solver-calculator/?f=%28lambda+-+3%29%2A%28lambda+-+1%29&var=lambda "equation solver")). These are the eigenvalues. Next, find the eigenvectors. - \$\$\$\\lambda = 3\$\$\$ \$\$\$\\left\[\\begin{array}{cc}1 - \\lambda & 2\\\\0 & 3 - \\lambda\\end{array}\\right\] = \\left\[\\begin{array}{cc}-2 & 2\\\\0 & 0\\end{array}\\right\]\$\$\$ The null space of this matrix is \$\$\$\\left\\{\\left\[\\begin{array}{c}1\\\\1\\end{array}\\right\]\\right\\}\$\$\$ (for steps, see [null space calculator](https://www.emathhelp.net/calculators/linear-algebra/null-space-calculator/?i=%5B%5B-2%2C2%5D%2C%5B0%2C0%5D%5D "null space calculator")). This is the eigenvector. - \$\$\$\\lambda = 1\$\$\$ \$\$\$\\left\[\\begin{array}{cc}1 - \\lambda & 2\\\\0 & 3 - \\lambda\\end{array}\\right\] = \\left\[\\begin{array}{cc}0 & 2\\\\0 & 2\\end{array}\\right\]\$\$\$ The null space of this matrix is \$\$\$\\left\\{\\left\[\\begin{array}{c}1\\\\0\\end{array}\\right\]\\right\\}\$\$\$ (for steps, see [null space calculator](https://www.emathhelp.net/calculators/linear-algebra/null-space-calculator/?i=%5B%5B0%2C2%5D%2C%5B0%2C2%5D%5D "null space calculator")). This is the eigenvector. Answer Eigenvalue: \$\$\$3\$\$\$A, multiplicity: \$\$\$1\$\$\$A, eigenvector: \$\$\$\\left\[\\begin{array}{c}1\\\\1\\end{array}\\right\]\$\$\$A. Eigenvalue: \$\$\$1\$\$\$A, multiplicity: \$\$\$1\$\$\$A, eigenvector: \$\$\$\\left\[\\begin{array}{c}1\\\\0\\end{array}\\right\]\$\$\$A.
Shard	63 (laksa)
Root Hash	16951690241190635063
Unparsed URL	net,emathhelp!www,/calculators/linear-algebra/eigenvalue-and-eigenvector-calculator/ s443