Inverse matrix

Algebra Level 4

If matrix $A=\begin{pmatrix} a & b \\ c & a \end{pmatrix}$ satisfies $A={A}^{-1},$ what are all possible values of real number $x$ such that the matrix $A-xI$ is not invertible?

Details and assumptions

$I$ denotes the identity matrix.

3, 4

1, -1

2, -2

1, 3

1 solution

Tom Engelsman
Dec 25, 2016

If $A = A^{-1}$ as stated above, then $AA^{-1} = I$ which produces:

$a^{2} + bc = 1; 2ab = 2ac = 0 \Rightarrow a = 0; bc = 1.$

If we wish to find a value $x$ such that $A - xI$ is non-invertible (or namely det $(A - xI) = 0$ ), then:

det $(A - xI)$ = $x^ 2 - 1 = 0 \Rightarrow \boxed{x = \pm1}.$

There are 2 solutions of 1st part one yrs and one b = c = 0 and $a^2 = 1$

Aniket Sanghi - 4 years, 3 months ago

Inverse matrix

1 solution

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