If A={bmatrix} 1 & -1 \\ -1 & 1 {bmatrix}, then the expression A³ - 2A² is

Correct Answer is : a null matrix

Correct Answer is : a null matrix

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UPSC NDA 2 2019 Mathematics Solved Paper

Question : 14 of 120

Marks: +1, -0

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

A^{3} - 2A^{2}

Solution:

Concept:

We compute powers of matrix

A

by matrix multiplication and then evaluate the given expression.

Explanation:

First, compute

A^2 = A \cdot A

A^2 = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} = \begin{bmatrix} 1\cdot 1+(-1)(-1) & 1\cdot (-1)+(-1)\cdot 1 \\ (-1)\cdot 1+1\cdot (-1) & (-1)(-1)+1\cdot 1 \end{bmatrix} = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix}

Next, compute

A^3 = A^2 \cdot A

A^3 = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} = \begin{bmatrix} 2\cdot 1+(-2)(-1) & 2\cdot (-1)+(-2)\cdot 1 \\ (-2)\cdot 1+2\cdot (-1) & (-2)(-1)+2\cdot 1 \end{bmatrix} = \begin{bmatrix} 4 & -4 \\ -4 & 4 \end{bmatrix}

Then compute

2A^2 = 2 \cdot \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} = \begin{bmatrix} 4 & -4 \\ -4 & 4 \end{bmatrix}

Finally,

A^3 - 2A^2 = \begin{bmatrix} 4 & -4 \\ -4 & 4 \end{bmatrix} - \begin{bmatrix} 4 & -4 \\ -4 & 4 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

, which is a null matrix.

Answer:

A null matrix.

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