Let A={bmatrix} x+y & y \\ 2x & x-y {bmatrix}, B={bmatrix} 2 \\ -1 {bmatrix} and C={bmatrix} 3 \\ 2 {bmatrix} If AB = C, then what is A² equal to? [NDA I 2015]

Correct Answer is : [6−10426]{bmatrix} 6 & -10 \\ 4 & 26 {bmatrix}[64−1026]

{bmatrix} 6 & -10 \\ 4 & 26 {bmatrix}

{bmatrix} -10 & 5 \\ 4 & 24 {bmatrix}

{bmatrix} -5 & -6 \\ -4 & -20 {bmatrix}

{bmatrix} -5 & -7 \\ -5 & 20 {bmatrix}

Correct Answer is : [6−10426]{bmatrix} 6 & -10 \\ 4 & 26 {bmatrix}[64−1026]

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NDA Math Matrices

Question : 60 of 69

Marks: +1, -0

Let

A=\begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix}, B=\begin{bmatrix} 2 \\ -1 \end{bmatrix}

C=\begin{bmatrix} 3 \\ 2 \end{bmatrix}

A^2

Solution:

Consider the following matrices

A=\begin{pmatrix} x+y & y \\ 2x & x-y \end{pmatrix}

B=\begin{pmatrix} 2 \\ -1 \end{pmatrix}

AB=\begin{pmatrix} x+y & y \\ 2x & x-y \end{pmatrix}\begin{pmatrix} 2 \\ -1 \end{pmatrix}

=\begin{pmatrix} 2x+2y-y \\ 4x-x+y \end{pmatrix}

=\begin{pmatrix} 2x+y \\ 3x+y \end{pmatrix}

=C

=\begin{pmatrix} \\ 3 \\ 2 \end{pmatrix}

Therefore, by comparing the elements of the matrices, we get

2x+y=3;\; 3x+y=2

Hence Solving for x and yields

x=-1,\; y=5

Therefore matrix A will be equal to

A=\begin{pmatrix} x+y & y \\ 2x & x-y \end{pmatrix}

=\begin{pmatrix} 4 & 5 \\ -2 & -6 \end{pmatrix}

Hence,the required answer is

A^2=\begin{pmatrix} 4 & 5 \\ -2 & -6 \end{pmatrix}\begin{pmatrix} 4 & 5 \\ -2 & -6 \end{pmatrix}

=\begin{pmatrix} 6 & -10 \\ 4 & 26 \end{pmatrix}

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