If and I is the unit matrix of order 2, then A2 equals

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

If $A = [\begin{array}{cc} 2 & - 1 \\ - 1 & 2 \end{array}]$ and I is the unit matrix of order 2, then A² equals

4A – 3I

3A – 4I

A – I

A + I

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$A = [\begin{array}{cc} 2 & - 1 \\ - 1 & 2 \end{array}]$

$A^{2} = A \times A = [\begin{array}{cc} 2 & - 1 \\ - 1 & 2 \end{array}] \times [\begin{array}{cc} 2 & - 1 \\ - 1 & 2 \end{array}]$

$= [\begin{array}{cc} 4 + 1 & - 2 - 2 \\ - 2 - 2 & 1 + 4 \end{array}] = [\begin{array}{cc} 5 & - 4 \\ - 4 & 5 \end{array}]$

From the given option to get first entry i.e 5 of A², only option (A) is satisfied. Or we can check as below

$4 A - 3 I = [\begin{array}{cc} 8 & - 4 \\ - 4 & 8 \end{array}] - 3 \times [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}]$

$= [\begin{array}{cc} 8 & - 4 \\ - 4 & 8 \end{array}] - [\begin{array}{cc} 3 & 0 \\ 0 & 3 \end{array}]$

$= [\begin{array}{cc} 5 & - 4 \\ - 4 & 5 \end{array}] = A^{2}$

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