If and , then a matrix Z such that X + Y + Z is a zero matrix.

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

If $X = [\begin{array}{ccc} 3 & 1 & - 1 \\ 5 & - 2 & - 3 \end{array}]$ and $Y = [\begin{array}{ccc} 2 & 1 & - 1 \\ 7 & 2 & 4 \end{array}]$ , then a matrix Z such that X + Y + Z is a zero matrix.

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We have, $X = {[\begin{array}{ccc} 3 & 1 & - 1 \\ 5 & - 2 & - 3 \end{array}]}_{2 \times 3}$ and $Y = {[\begin{array}{ccc} 2 & 1 & - 1 \\ 7 & 2 & 4 \end{array}]}_{2 \times 3}$

$X + Y = [\begin{array}{ccc} 3 + 2 & 1 + 1 & - 1 - 1 \\ 5 + 7 & - 2 + 2 & - 3 + 4 \end{array}]$

$= [\begin{array}{ccc} 5 & 2 & - 2 \\ 12 & 0 & + 1 \end{array}]$

Also, $X + Y + Z = [\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]$

We see that Z is the additive inverse of (x + y) or negative of (x + y)

$∴ Z = [\begin{array}{ccc} - 5 & - 2 & 2 \\ - 12 & 0 & - 1 \end{array}]$

[∵ Z = – (X + Y)]

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