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If $x, y, z$ are nonzero real numbers, then the inverse of matrix $A=\left[\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]$ is (A) $\left[\begin{array}{ccc}x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1}\end{array}\right]$ (B) $x y z\left[\begin{array}{ccc}x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1}\end{array}\right]$ (C) $\frac{1}{x y z}\left[\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]$ (D) $\frac{1}{x y z}\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

Name: Problem 18, Chapter 4: NCERT Class 12 Part 1 - Math
Uploaded: 2021-06-13T03:37:37-08:00
Duration: 16 s
Channel: Tanishq Gupta
Description: If x, y, z are nonzero real numbers, then the inverse of matrix A=[ x 0 0 0 y 0 0 0 z ] is (A) [ x^-1 0 0 0 y^-1 0 0 0 z^-1 ] (B) x y z[ x^-1 0 0 0 y^-1 0 0 0 z^-1 ] (C) (1)/(x y z)[ x 0 0 0 y 0 0 0 z ] (D) (1)/(x y z)[ 1 0 0 0 1 0 0 0 1 ]

   If $x, y, z$ are nonzero real numbers, then the inverse of matrix $A=\left[\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]$ is
(A) $\left[\begin{array}{ccc}x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1}\end{array}\right]$
(B) $x y z\left[\begin{array}{ccc}x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1}\end{array}\right]$
(C) $\frac{1}{x y z}\left[\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]$
(D) $\frac{1}{x y z}\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

NCERT Class 12 Part 1 - Math

Pawan K. Jain 2005 Edition

Chapter 4, Problem 18

Instant Answer

Solved by Expert Tanishq Gupta

06/13/2021

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The inverse of a diagonal matrix is obtained by taking the reciprocal of each of the diagonal elements. Show more…

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If $x, y, z$ are nonzero real numbers, then the inverse of matrix $A=\left[\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]$ is (A) $\left[\begin{array}{ccc}x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1}\end{array}\right]$ (B) $x y z\left[\begin{array}{ccc}x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1}\end{array}\right]$ (C) $\frac{1}{x y z}\left[\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]$ (D) $\frac{1}{x y z}\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

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Key Concepts

Matrix Inversion

Matrix inversion is the process of finding a matrix that, when multiplied by the original matrix, yields the identity matrix. This key concept is central in solving systems of linear equations and understanding linear transformations in linear algebra.

Diagonal Matrix

A diagonal matrix is a matrix where all off-diagonal entries are zero. Its structure simplifies many operations, including matrix inversion, because each diagonal element can be manipulated independently.

Multiplicative Inverse

The multiplicative inverse of a number is its reciprocal, which when multiplied by the original number yields one. In the context of diagonal matrices, each nonzero diagonal element is replaced by its multiplicative inverse to obtain the inverse of the matrix.

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