Inverse of the matrix [ p 0 0 0 q 0 0 0 r ] where pqr ≠0 is

Inverse of the matrix [ p 0 0 0 q 0 ...

Inverse of the matrix $\left[\begin{array}{lll}\mathrm{p} & 0 & 0 \\ 0 & \mathrm{q} & 0 \\ 0 & 0 & \mathrm{r}\end{array}\right]$ where pqr $\neq 0$ is (a) $\left[\begin{array}{lll}0 & 0 & \mathrm{p} \\ 0 & \mathrm{q} & 0 \\ \mathrm{r} & 0 & 0\end{array}\right]$ (b) $\left[\begin{array}{lll}\mathrm{p} & 0 & 0 \\ 0 & \mathrm{q} & 0 \\ 0 & 0 & \mathrm{r}\end{array}\right]$ (c) $\left[\begin{array}{ccc}0 & 0 & -\mathrm{p} \\ 0 & -\mathrm{q} & 0 \\ -\mathrm{r} & 0 & 0\end{array}\right]$ (d) $\left[\begin{array}{ccc}p^{-1} & 0 & 0 \\ 0 & q^{-1} & 0 \\ 0 & 0 & r^{-1}\end{array}\right]$

Inverse of the matrix $\left[\begin{array}{lll}\mathrm{p} & 0 & 0 \\ 0 & \mathrm{q} & 0 \\ 0 & 0 & \mathrm{r}\end{array}\right]$ where pqr $\neq 0$ is
(a) $\left[\begin{array}{lll}0 & 0 & \mathrm{p} \\ 0 & \mathrm{q} & 0 \\ \mathrm{r} & 0 & 0\end{array}\right]$
(b) $\left[\begin{array}{lll}\mathrm{p} & 0 & 0 \\ 0 & \mathrm{q} & 0 \\ 0 & 0 & \mathrm{r}\end{array}\right]$
(c) $\left[\begin{array}{ccc}0 & 0 & -\mathrm{p} \\ 0 & -\mathrm{q} & 0 \\ -\mathrm{r} & 0 & 0\end{array}\right]$
(d) $\left[\begin{array}{ccc}p^{-1} & 0 & 0 \\ 0 & q^{-1} & 0 \\ 0 & 0 & r^{-1}\end{array}\right]$

Key Concepts

Diagonal Matrix

A diagonal matrix is a matrix in which all off-diagonal entries are zero. This property simplifies many matrix operations because each element on the main diagonal can be handled independently from the others.

Matrix Inverse

The inverse of a matrix A, denoted A?¹, is defined such that when it is multiplied by A, the result is the identity matrix. For a matrix to have an inverse, it must be square and have a non-zero determinant.

Inversion of Diagonal Matrices

For a diagonal matrix with nonzero elements on its main diagonal, the inversion process is straightforward. The inverse is found by simply taking the reciprocal of each of the diagonal entries, resulting in another diagonal matrix where each diagonal element is the reciprocal of the original.

Transcript

00:01 So in the given question we are told to find the inverse of a matrix that is given as p 0 0 q 0 0 0 0 0 0 0 and we are told that p times q times r is not equal to 0.

00:21 So let's say this matrix is the matrix a.

00:25 So we can find the determinant of the given matrix as determinant of a would be equal to p times q times r.

00:43 So in order to find the inverse of the matrix a, what we should need is the inverse of a matrix is equal to a inverse is equal to a inverse is equal.

00:59 To the adjoint matrix of a divided by the determinant of a and in order to find the adjoint matrix of a we need to find the adjoint of a is equal to the transpose of the co -factor matrix and in a co -factor matrix let's say we have the we have to find the co -factor matrix for the given matrix a.

01:34 So in the cofactor matrix the first element or we can just generalize this result, let's say the element in the ith row and the jeth column is found by minus 1 raised to i plus j times the minor of the element at that position.

01:57 Right so for example if you are taking the element p that is in the first row and the first column we have i and j as 1 1 and c1 would be equal to minus 1 raised to 1 plus 1 and the minor of p is the small matrix the determinant of the small matrix that comes over here right so this would be equal to q times r, right? the determining is q times r minus 0, which is 2 times r, minus 1 raised to 1 plus 1 is minus 1 squared, which is equal to 1.

02:41 So c11 is qr, right? now similarly we can find the rest of the matrix rest of the elements of the matrix also...

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