# What is meant by the determinant of a matrix?

Jul 1, 2018

Assuming that we have a square matrix, then the determinant of the matrix is the determinant with the same elements.

Eg if we have a $2 \times 2$ matrix:

$\boldsymbol{A} = \left(\begin{matrix}a & b \\ c & d\end{matrix}\right)$

The the associated determinant given by

$D = | \boldsymbol{A} | = | \left(a , b\right) , \left(c , d\right) | = a d - b c$

Jul 1, 2018

See below.

#### Explanation:

To extend on Steve's explanation, the determinant of a matrix tells you whether or not the matrix is invertible. If the determinant is 0, the matrix is not invertible.

For example, let $A = \left(\begin{matrix}1 & 3 \\ - 2 & 1\end{matrix}\right)$. Then $\det \left(A\right) = 1 \left(1\right) - 3 \left(- 2\right) = 7$ so we know that ${A}^{-} 1$ exists.
If we let $B = \left(\begin{matrix}1 & 2 \\ - 2 & - 4\end{matrix}\right)$, $\det \left(B\right) = 1 \left(- 4\right) - 2 \left(- 2\right) = 0$ so we know that ${B}^{-} 1$ doesn't exist.
Additionally, the determinant is involved in computing the inverse of a matrix. Given a matrix $A = \left(\begin{matrix}a & b \\ c & d\end{matrix}\right)$, ${A}^{-} 1 = \frac{1}{\det} \left(A\right) \left(\begin{matrix}d & - b \\ - c & a\end{matrix}\right)$. From this, you can see why ${A}^{-} 1$ doesn't exist when $\det \left(A\right) = 0$.

Jul 1, 2018

Also area / volume scale factor...

#### Explanation:

The determinant is also used as a area/volume scale factor,

If we have a $2 \times 2$ matrix, $M$

Then if a particular shape of area $A$ undergoes the transformation defined by the matrix $M$ then the area of the new shape will be $\det \left(M\right) A$ or $| M | A$

Also

$\det \left(M\right) = 0 \iff \text{ M defined as being 'singular' , no inverse}$