First slide

Applications of determinant

Question

The inverse of a diagonal matrix is

Moderate

A

a diagonal matrix
B

a skew –symmetric matrix
C

a symmetric matrix
D

none of these

Solution

$\begin{array}{l} A = diag (d_{1}, d_{2}, d_{3}, \dots \dots d_{n}) \\ \Rightarrow | A | = (d_{1} \times d_{2} \times d_{3} \times d_{4} \dots \dots \dots . d_{n}) \end{array}$
Now,
Cofactor of $d_{1} i s d_{2} d_{3} .... d_{n}$
Cofactor of $d_{2} i s d_{1} \times d_{3} \times d_{4} ...... d_{n}$
Cofactor of $d_{3} i s d_{1} \times d_{2} \times d_{4} ...... d_{n}$
$Cofactor of d_{n} is d_{1} \times d_{2} \times d_{3} \dots \dots d_{n - 1}$
$\Rightarrow A^{- 1} = \frac{1}{| A |} a d j (A) = d i a g ({d_{1}}^{- 1}, d_{2}^{- 1}, d_{3}^{- 1}, ...... d_{n}^{- 1})$
$Hence A^{- 1} is also a diagonal matrix.$

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