Question

Inverse of diagonal matrix (if it exists) is a

• A. Skew-symmetric matrix
• B. Diagonal matrix
• C. Non invertible matrix
• D. None of these

Answer 1

Answer: (B)

Diagonal matrix

Answer 2

Let $A = \text{diag}(d_{1},d_{2},d_{3}......,d_{n})$

As A is invertible, therefore $\det(A) \neq 0$

⇒ $d_{1},d_{2},d_{3}.......,d_{n} \neq 0$ ⇒ $d_{i} \neq 0$ for i = 1, 2, 3…..n

Here, cofactor of each non diagonal entry is 0 and cofactor of $a_{ii}$

$$$.d_{i + 1},......,d_{n}$ $= \frac { 1 } { d _ { i } } \left[ d _ { 1 } , d _ { 2 } , d _ { 3 } \ldots . , d _ { i - 1 } , d _ { i } , d _ { i + 1 } \ldots . . , d _ { n } \right] = \frac { | A | } { d _ { i } }$ $A^{- 1} = \frac{1}{|A|}(adjA) = diag\left( \frac{1}{d_{1}},\frac{1}{d_{2}}.......,\frac{1}{d_{n}} \right),$ which is a diagonal matrix