If and , then show that A2

Engineering

Mathematics

Algebra of Matrices

Multiplication of Matrices

Special Type of Square Matrices

Question

If $A = [\begin{array}{cc} 3 & 1 \\ - 1 & 2 \end{array}]$ and $I = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}]$ , then show that A² – 5A + 7I₂ = 0

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A = [\begin{array}{cc} 3 & 1 \\ - 1 & 2 \end{array}] I = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] A^{2} = 5 A + 7 I_{2} = 0

A . A = [\begin{array}{cc} - 3 & 1 \\ - 1 & 2 \end{array}] [\begin{array}{cc} - 3 & 1 \\ - 1 & 2 \end{array}] = [\begin{array}{cc} 9 - 1 & + 3 - 2 \\ 63 + 2 & - 1 + 4 \end{array}] = [\begin{array}{cc} 8 & + 5 \\ 5 & 3 \end{array}]

= [\begin{array}{cc} 8 & + 5 \\ 5 & 3 \end{array}] - [\begin{array}{cc} 15 & 5 \\ - 5 & 10 \end{array}] + [\begin{array}{cc} 7 & 0 \\ 0 & 7 \end{array}]

= [\begin{array}{cc} 15 & + 5 \\ 5 & 10 \end{array}] - [\begin{array}{cc} 15 & 5 \\ 5 & 10 \end{array}] = 0