First slide

Applications of determinant

Question

If $[\begin{array}{ll} 2 1 \\ 7 4 \end{array}] A [\begin{array}{rr} - 3 2 \\ 5 - 3 \end{array}] = [\begin{array}{ll} 1 0 \\ 0 1 \end{array}]$ , then matrix A equals

Moderate

A

$[\begin{array}{cc} 7 & 5 \\ - 11 & - 8 \end{array}]$
B

$[\begin{array}{ll} 2 1 \\ 5 3 \end{array}]$
C

$[\begin{array}{cc} 7 & 1 \\ 34 & 5 \end{array}]$
D

$[\begin{array}{cc} 5 & 3 \\ 13 & 8 \end{array}]$

Solution

If XAY=I, then $A = X^{- 1} Y^{- 1} = (Y X)^{- 1}$
In this case $Y X = [\begin{array}{cc} - 3 & 2 \\ 5 & - 3 \end{array}] [\begin{array}{ll} 2 1 \\ 7 4 \end{array}] = [\begin{array}{cc} 8 & 5 \\ - 11 & - 7 \end{array}]$
$∴ A = {[\begin{array}{cc} 8 & 5 \\ - 11 & - 7 \end{array}]}^{- 1} = [\begin{array}{cc} 7 & 5 \\ - 11 & - 8 \end{array}]$

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