First slide

Applications of determinant

Question

Let $A = [\begin{array}{ll} a b \\ c d \end{array}], a, b, c, d \in R$ If $A^{5} = A^{3} + I$ , then A is

Moderate

A

a symmetric matrix
B

a skew symmetric matrix
C

an invertible matrix
D

none of these

Solution

$A^{5} = A^{3} + I$
$\Rightarrow A (A^{4} - A^{2}) = (A^{4} - A^{2}) A = I$
$\Rightarrow$ A is invertible and $A^{- 1} = A^{4} - A^{2}$ .

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