### Matrix calculus: Determinants

### Row and column expansion

We will now focus on the actual calculation of determinants. There is a range of useful mathematical rules. We will focus here on the so-called expansion of a determinant along a row or column. Below we discuss the role of row and column operations on matrices.

The gist of expansion along a row or column is that the calculation of a determinant is reduced to a calculation of smaller determinants. Here is the exact wording.

Expansion along a row or column

The determinant of an #(n\times n)#-matrix #A# can be calculated by expansion along a row or column. By this we mean the following equations, where #A_{ij}# is the #((n-1)\times(n-1))#-matrix obtained from #A# by deleting the #i#-th row and the #j#-th column.

- Expansion along the #i#-th row:

\[

\det (A)=\sum_{j=1}^n (-1)^{i+j}a_{ij}\det (A_{ij})

\] - Expansion along the #j#-th column:

\[

\det (A)=\sum_{i=1}^n (-1)^{i+j}a_{ij}\det (A_{ij})

\]

Let \(A\) be the following \((3\times3)\)-matrix: \[A=\matrix{2 & -4 & 4 \\ -5 & 0 & -5 \\ 4 & -5 & 2 \\ }\]

Calculate the determinant of #A# by expansion along a row or column.

Calculate the determinant of #A# by expansion along a row or column.

Solution \(\det (A)= \) \(90\)

The # (2,2)#-entry of #A# equals #0#. Therefore we expand along the second row.

\[\begin{array}{rcl}\det(A) &=& (-1)^{2+1}\cdot a_{2 1}\cdot \det(A_{2 1}) + (-1)^{2+2}\cdot a_{2 2}\cdot \det(A_{2 2}) + (-1)^{2+3}\cdot a_{2 3}\cdot \det(A_{2 3})\\

&=& 5\cdot\left|\begin{array}{cc}-4&4\\-5&2\end{array}\right| +0\cdot\left|\begin{array}{cc}2&4\\4&2\end{array}\right| +5\cdot\left|\begin{array}{cc}2&-4\\4&-5\end{array}\right|\\

&=& 5\cdot12 +0 +5\cdot6\\

&=& 90

\end{array}\]

The # (2,2)#-entry of #A# equals #0#. Therefore we expand along the second row.

\[\begin{array}{rcl}\det(A) &=& (-1)^{2+1}\cdot a_{2 1}\cdot \det(A_{2 1}) + (-1)^{2+2}\cdot a_{2 2}\cdot \det(A_{2 2}) + (-1)^{2+3}\cdot a_{2 3}\cdot \det(A_{2 3})\\

&=& 5\cdot\left|\begin{array}{cc}-4&4\\-5&2\end{array}\right| +0\cdot\left|\begin{array}{cc}2&4\\4&2\end{array}\right| +5\cdot\left|\begin{array}{cc}2&-4\\4&-5\end{array}\right|\\

&=& 5\cdot12 +0 +5\cdot6\\

&=& 90

\end{array}\]

Unlock full access

Teacher access

Request a demo account. We will help you get started with our digital learning environment.

Student access

Is your university not a partner?
Get access to our courses via

Or visit omptest.org if jou are taking an OMPT exam.

**Pass Your Math**independent of your university. See pricing and more.Or visit omptest.org if jou are taking an OMPT exam.