Practice: Expressing Linear Equations As Matrices

This page contains practice exercises for expressing linear equations as matrices.

1\. Express the following set of linear equations as a matrix.

2 x + 4 y = 6 2 x + 6 y = 8 {\displaystyle {\begin{aligned}2x+4y=6\\\2x+6y=8\end{aligned}}}

solution

[ 2 4 2 6 ] [ x y ] = [ 6 8 ] {\displaystyle {\begin{bmatrix}2&4\\\2&6\end{bmatrix}}{\begin{bmatrix}x\\\y\end{bmatrix}}={\begin{bmatrix}6\\\8\end{bmatrix}}}

3\. Express the following set of linear equations as a matrix

4 a − b = 4 5 a − 2 b = 2 a + 3 b = 4 {\displaystyle {\begin{aligned}4a-b=4\\\5a-2b=2\\\a+3b=4\end{aligned}}}

solution

[ 4 − 1 5 − 2 1 3 ] [ a b ] = [ 4 2 4 ] {\displaystyle {\begin{bmatrix}4&-1\\\5&-2\\\1&3\end{bmatrix}}{\begin{bmatrix}a\\\b\end{bmatrix}}={\begin{bmatrix}4\\\2\\\4\end{bmatrix}}}

2\. Express the following set of linear equations as a matrix

2 x 1 + x 2 + 7 x 3 = 5 x 1 + x 2 + 5 x 3 = 3 {\displaystyle {\begin{aligned}2x_{1}+x_{2}+7x_{3}=5\\\x_{1}+x_{2}+5x_{3}=3\end{aligned}}}

solution

[ 2 1 7 1 1 5 ] [ x 1 x 2 x 3 ] = [ 5 3 ] {\displaystyle {\begin{bmatrix}2&1&7\\\1&1&5\end{bmatrix}}{\begin{bmatrix}x_{1}\\\x_{2}\\\x_{3}\end{bmatrix}}={\begin{bmatrix}5\\\3\end{bmatrix}}}

4\. Express the following set of linear equations as a matrix

3 r − s + t = 10 2 r + 4 t = 11 15 r = 0 {\displaystyle {\begin{aligned}3r-s+t=10\\\2r+4t=11\\\15r=0\end{aligned}}}

solution

[ 3 − 1 1 2 0 4 15 0 0 ] [ r s t ] = [ 10 11 0 ] {\displaystyle {\begin{bmatrix}3&-1&1\\\2&0&4\\\15&0&0\end{bmatrix}}{\begin{bmatrix}r\\\s\\\t\end{bmatrix}}={\begin{bmatrix}10\\\11\\\0\end{bmatrix}}}

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