If $A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}$, $B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}$ and $(A+B)^2 = A^2 + B^2$, then the values of $a$ and $b$ are: (a) a = 4, b = 1 (b) a = 1, b = 4 (c) a = 0, b = 4 (d) a = 2, b = 4

If A = 1 -1 \2 -1 , B = a 1 b -1 and (A+B)² = A² + B², then the values of a and b are: (a) a = 4, b = 1 (b) a

Question 92 of 110

A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}

(a) a = 4, b = 1

(b) a = 1, b = 4

(d) a = 2, b = 4

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$(A+B)^2=A^2+B^2$ requires $AB+BA=0$ (a zero matrix); computing this from $A,B$ gives $a=1,\,b=4$ .

Expanding, $(A+B)^2=A^2+AB+BA+B^2$ . For this to equal $A^2+B^2$ , we need $AB+BA=0$ (the zero matrix), which is the real condition to use (matrix multiplication is not commutative, so we cannot cancel $AB$ against $BA$ individually).

With $A=\begin{bmatrix}1&-1\\2&-1\end{bmatrix}$ , $B=\begin{bmatrix}a&1\\b&-1\end{bmatrix}$ :

$AB=\begin{bmatrix}a-b&2\\2a-b&3\end{bmatrix},\qquad BA=\begin{bmatrix}a+2&-a-1\\b-2&1-b\end{bmatrix}.$ …

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Q.If $A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}$ , $B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}$ and $(A+B)^2 = A^2 + B^2$ , then the values of $a$ and $b$ are:

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

Q.If A=[1−12−1]A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}A=[12​−1−1​], B=[a1b−1]B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}B=[ab​1−1​] and (A+B)2=A2+B2(A+B)^2 = A^2 + B^2(A+B)2=A2+B2, then the values of aaa and bbb are:

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Q.If $A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}$ , $B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}$ and $(A+B)^2 = A^2 + B^2$ , then the values of $a$ and $b$ are:

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