If $A = \begin{bmatrix} \lambda & 1 \\ -1 & -\lambda \end{bmatrix}$, then for what value of $\lambda$, $A^2 = O$? (1) $0$ (2) $\pm 1$ (3) $-1$ (4) $1$

If A = λ 1 \-1 -λ , then for what value of λ, A² = O? (1) 0 (2) ± 1 (3) -1 (4) 1

Exercise 7.5 · Q5

A = \begin{bmatrix} \lambda & 1 \\ -1 & -\lambda \end{bmatrix}

(1)

0

(2)

\pm 1

(3)

-1

(4)

1

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Square the matrix and equate to $O$ .

This tests matrix multiplication and solving for a parameter.

Step 1. With $A = \begin{bmatrix} \lambda & 1 \\ -1 & -\lambda \end{bmatrix}$ , compute $A^2$ . Top-left: $\lambda\cdot\lambda + 1\cdot(-1) = \lambda^2 - 1$ . Top-right: $\lambda\cdot1 + 1\cdot(-\lambda) = 0$ . …

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Q.If $A = \begin{bmatrix} \lambda & 1 \\ -1 & -\lambda \end{bmatrix}$ , then for what value of $\lambda$ , $A^2 = O$ ?

Concept understanding — Properties of Matrix Addition, Scalar Multiplication and Product of Matrices

Q.If A=[λ1−1−λ]A = \begin{bmatrix} \lambda & 1 \\ -1 & -\lambda \end{bmatrix}A=[λ−1​1−λ​], then for what value of λ\lambdaλ, A2=OA^2 = OA2=O?

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Q.If $A = \begin{bmatrix} \lambda & 1 \\ -1 & -\lambda \end{bmatrix}$ , then for what value of $\lambda$ , $A^2 = O$ ?

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