{"id":4996,"date":"2021-09-04T00:18:35","date_gmt":"2021-09-03T18:48:35","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4996"},"modified":"2022-01-16T17:01:49","modified_gmt":"2022-01-16T11:31:49","slug":"how-to-find-trace-of-matrix","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/how-to-find-trace-of-matrix\/","title":{"rendered":"How to Find Trace of Matrix – Properties and Example"},"content":{"rendered":"

Here you will learn how to find trace of matrix, its properties and what is orthogonal matrix with example.<\/p>\n

Let’s begin –<\/p>\n

Trace of Matrix<\/h2>\n

The sum of the elements of the square matrix A lying along the principal diagonal<\/strong> is called the trace of A<\/strong> i.e (tr(A))<\/strong>. <\/p>\n

\n

Thus if  A = \\([a_{ij}]_{n\\times n}\\),<\/p>\n

then tr(A) = \\(\\sum_{i=1}^{n}\\) \\(a_{ii}\\) = \\(a_{11}\\) + \\(a_{22}\\) + ……… + \\(a_{nn}\\).<\/p>\n<\/blockquote>\n

How to Find Trace of Matrix :<\/strong><\/p>\n

for example, for 3×3 matrix<\/strong>, if A = \\(\\begin{bmatrix} 2 & 1 & -1 \\\\ 3 &  -2 & 5 \\\\ 1 & 5 &  3  \\end{bmatrix}\\) <\/p>\n

then, trace of A <\/strong>or tr(A)<\/strong> = 2 + (-2) + 3 = 3<\/p>\n

for example, for 2×2 matrix<\/strong>, if A = \\(\\begin{bmatrix} 2 & 1 \\\\ 3 &  4  \\end{bmatrix}\\) <\/p>\n

then, trace of A <\/strong>or tr(A)<\/strong> = 2 + 4 = 6<\/p>\n

Properties of Trace of a Matrix<\/strong><\/h4>\n

Let A = \\([a_{ij}]_{n\\times n}\\) and B = \\([b_{ij}]_{n\\times n}\\) and \\(\\lambda\\) be a scalar then<\/p>\n

\n

(i)  tr(\\(\\lambda A\\))  = \\(\\lambda\\) tr(A) <\/p>\n

(ii) tr(A + B) = tr(A) + tr(B)<\/p>\n

(iii) tr(AB) = tr(BA)<\/p>\n<\/blockquote>\n

Orthogonal Matrix<\/h2>\n

A square matrix is said to be orthogonal matrix if <\/p>\n

\n

 \\(AA^T\\) = I (Identity matrix)<\/p>\n<\/blockquote>\n

Note<\/strong> :  The determinant value of orthgonal matrix is 1 or -1.<\/p>\n\n\n

Example : <\/span>Show that the matrix A = \\(\\begin{bmatrix} cosx & sinx \\\\ -sinx &  cosx  \\end{bmatrix}\\) is a orthogonal matrix.<\/p>\n

Solution : <\/span>We have,

\nA = \\(\\begin{bmatrix} cosx & sinx \\\\ -sinx &  cosx  \\end{bmatrix}\\)

\n\\(A^{T}\\) = \\(\\begin{bmatrix} cosx & -sinx \\\\ sinx &  cosx  \\end{bmatrix}\\)

\nNow, we have to find \\(AA^T\\) = \\(\\begin{bmatrix} cosx & sinx \\\\ -sinx &  cosx  \\end{bmatrix}\\)\\(\\begin{bmatrix} cosx & -sinx \\\\ sinx &  cosx  \\end{bmatrix}\\)

\n= \\(\\begin{bmatrix} cos^2x + sin^2x & -cosx.sinx + sinx.cosx \\\\ -sinx.cosx + sinx.cosx &  cos^2x + sin^2x  \\end{bmatrix}\\)

\n= \\(\\begin{bmatrix} 1 & 0 \\\\ 0 & 1  \\end{bmatrix}\\) = I (Identity matrix)

\n\\(\\implies\\) \\(AA^T\\) = I

\nHence, it is an orthogonal matrix.

<\/p>\n\n\n\n

\n
Next – Adjoint of the Matrix (2\u00d72 & 3\u00d73) \u2013 Properties, Examples<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Symmetric and Skew Symmetric Matrices<\/a><\/div>\n<\/div>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find trace of matrix, its properties and what is orthogonal matrix with example. Let’s begin – Trace of Matrix The sum of the elements of the square matrix A lying along the principal diagonal is called the trace of A i.e (tr(A)).  Thus if  A = \\([a_{ij}]_{n\\times n}\\), then …<\/p>\n

How to Find Trace of Matrix – Properties and Example<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[34],"tags":[478,476,477],"yoast_head":"\nHow to Find Trace of Matrix - Properties and Example<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find trace of matrix, its properties and what is orthogonal matrix with example.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/how-to-find-trace-of-matrix\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How to Find Trace of Matrix - 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