Question

Create a matrix for this linear system: 3x+2y+z = 26 X-4y = -11 2x+z = 13 The determinant of the coefficient matrix is _______. x = y = z =

Answer 1

`\mathrm{Matrix \: Form:}`

`\begin{pmatrix}3&2&1\\ 1&-4&0\\ 2&0&1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix} = \begin{pmatrix}26\\ -11\\ 13\end{pmatrix}`

`\mathrm{Determinant\; of \;coefficient \;matrix = - 6}`

Explanation:

The system of linear equations provided in the question:

`\begin{aligned}3x+2y+z &= 26\\ x-4y&=-11\\ 2x+z&=13\\\\\end{aligned}`

can be represented in matrix form as

`\begin{pmatrix}3&2&1\\ 1&-4&0\\ 2&0&1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix} = \begin{pmatrix}26\\ -11\\ 13\end{pmatrix}`

The coefficient matrix is

`\begin{pmatrix}3&2&1\\ 1&-4&0\\ 2&0&1\end{pmatrix}`

The determinant of this matrix can be obtained by the expansion formula:

`\det \begin{pmatrix}a&b&c\\ \:d&e&f\\ \:g&h&i\end{pmatrix}=a\cdot \det \begin{pmatrix}e&f\\ \:h&i\end{pmatrix}-b\cdot \det \begin{pmatrix}d&f\\ \:g&i\end{pmatrix}+c\cdot \det \begin{pmatrix}d&e\\ \:g&h\end{pmatrix}`

Therefore

`det \begin{pmatrix}3&2&1\\ 1&-4&0\\ 2&0&1\end{pmatrix}`

`= 3\cdot \det \begin{pmatrix}-4&0\\ 0&1\end{pmatrix}-2\cdot \det \begin{pmatrix}1&0\\ 2&1\end{pmatrix}+1\cdot \det \begin{pmatrix}1&-4\\ 2&0\end{pmatrix}`

`\mathrm{Find\:the\:matrix\:determinant\:according\:to\:formula}:\quad \det \begin{pmatrix}a\:&\:b\:\\ c\:&\:d\:\end{pmatrix}\:=\:ad-bc`

`\det \begin{pmatrix}-4&0\\ 0&1\end{pmatrix} = \left(-4\right)\cdot \:1-0\cdot \:0 = -4`

`\det \begin{pmatrix}1&0\\ 2&1\end{pmatrix} = 1\cdot \:1-0\cdot \:2 = 1`

`\det \begin{pmatrix}1&-4\\ 2&0\end{pmatrix} = 1\cdot \:0-\left(-4\right)\cdot \:2 =8`

Finally,

`det \begin{pmatrix}3&2&1\\ 1&-4&0\\ 2&0&1\end{pmatrix}\\\\\\ = 3\cdot \det \begin{pmatrix}-4&0\\ 0&1\end{pmatrix}-2\cdot \det \begin{pmatrix}1&0\\ 2&1\end{pmatrix}+1\cdot \det \begin{pmatrix}1&-4\\ 2&0\end{pmatrix}\\\\\\= 3\left(-4\right)-2\cdot \:1+1\cdot \:8\\\\= -12 - 2 + 8\\\\= -6`