If , find k so that A2 = kA

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

If $A = [\begin{array}{ll} 3 - 2 \\ 4 - 2 \end{array}]$ , find k so that A² = kA – 2I

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$A = [\begin{array}{ll} 3 - 2 \\ 4 - 2 \end{array}]$

$A^{2} = [\begin{array}{ll} 3 - 2 \\ 4 - 2 \end{array}] [\begin{array}{ll} 3 - 2 \\ 4 - 2 \end{array}] = [\begin{array}{ll} 1 - 2 \\ 4 - 4 \end{array}]$

$kA = [\begin{array}{ll} 3 k - 2 k \\ 4 k - 2 k \end{array}], 2 I = [\begin{array}{ll} 2 0 \\ 0 2 \end{array}]$

kA – A² = 2I

$[\begin{array}{ll} 3 k - 2 k \\ 4 k - 2 k \end{array}] - [\begin{array}{ll} 1 - 2 \\ 4 - 4 \end{array}] = [\begin{array}{ll} 2 0 \\ 0 2 \end{array}]$

$[\begin{array}{cc} 3 k - 1 & - 2 k + 2 \\ 4 k - 4 & - 2 k + 4 \end{array}] = [\begin{array}{ll} 2 0 \\ 0 2 \end{array}]$

on comparing the elements of matrices on LHS and RHs, we get

k = 1

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