If , show that .

Engineering

Mathematics

Algebra of Matrices

Multiplication of Matrices

Special Type of Square Matrices

Question

If $A = [\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}] [\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}]$ , show that $A^{2} = [\begin{array}{cc} 1 & 2 \\ 0 & 1 \end{array}] [\begin{array}{cc} 1 & 3 \\ 0 & 1 \end{array}]$ .

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$A = [\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}]$

$A^{2} = [\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}] [\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}]$

$A^{2} = [\begin{array}{cc} 1 + 0 & 1 + 1 \\ 0 + 0 & 0 + 1 \end{array}]$

$∴ A^{2} = [\begin{array}{cc} 1 & 2 \\ 0 & 1 \end{array}]$

Now, $A^{3} = A^{2} . A = [\begin{array}{cc} 1 & 2 \\ 0 & 1 \end{array}] [\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}]$

$A^{3} = [\begin{array}{cc} 1 + 0 & 1 + 2 \\ 0 + 0 & 0 + 1 \end{array}]$

$∴ A^{3} = [\begin{array}{cc} 1 & 3 \\ 0 & 1 \end{array}]$

Hence proved.