Determinants of Matrices (Edexcel A Level Further Maths: Core Pure): Revision Note

Determinant of a 2x2 matrix

What is a determinant?

• The determinant is a numerical value (positive or negative) calculated from the elements in a matrix and is used to find the inverse of a matrix

• You can only find the determinant of a square matrix

• The method for finding the determinant of a $2\times 2$ matrix is given by: 

$\mathit{A}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\Rightarrow \det \mathit{A}=\left|\mathit{A}\right|=ad-bc$

Determinant of a 3x3 matrix

What is the minor  of an element in a 3x3 matrix?

• For any element in a 3x3 matrix, the minor is the determinant of the 2x2 matrix created by crossing out the row and column containing that element

• For the matrix $\left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)$

  • The minor of the element a would be found by:

    • crossing out the first row and first column $\left(\begin{array}{ccc}\sout{\sout{a}}& \sout{b}& \sout{c}\\ \overline{)d}& e& f\\ \overline{)g}& h& i\end{array}\right)$ 

    • finding the determinant of the remaining 2x2 matrix $\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|=ei-fh$

  • The minor of the element f would be found by:

    • crossing out the second row and third column $\left(\begin{array}{ccc}a& b& \overline{)c}\\ \sout{d}& \sout{e}& \sout{f}\\ g& h& \overline{)i}\end{array}\right)$

    • finding the determinant of the remaining 2x2 matrix $\left|\begin{array}{cc}a& b\\ g& h\end{array}\right|=ah-bg$

How do I find the determinant of a 3x3 matrix?

• Finding the determinant of a 3x3 matrix is best explained using an example

• STEP 1 Select any row or column in the matrix

  • e.g.  Selecting Row 2 of $\mathit{M}=\left(\begin{array}{ccc}4& -2& 3\\ 5& 6& -7\\ -8& 2& -1\end{array}\right)$

• STEP 2 Use the matrix of signs $\left(\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right)$ to find the row or column that corresponds to the row or column selected in Step 1

  • e.g. $\left(\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right)$ Row 2 was selected so “-  +  -“ will be needed

• STEP 3 Multiply each element in the selected row or column by its minor and use the corresponding signs form the matrix of signs to determine whether to add or subtract each product

  • e.g. $\det \mathit{M}=-\left(5\right)\left|\begin{array}{cc}-2& 3\\ 2& -1\end{array}\right|+\left(6\right)\left|\begin{array}{cc}4& 3\\ -8& -1\end{array}\right|-(-7)\left|\begin{array}{cc}4& -2\\ -8& 2\end{array}\right|$

• STEP 4 Evaluate the minors and calculate the determinant

  • e.g. $\det \mathit{M}=-\left(5\right)(-4)+\left(6\right)\left(20\right)-(-7)(-8)=84$

  • Check the answer using a calculator with a matrix feature

Properties of determinants

What are the properties of determinants of matrices?

• The determinant of an identity matrix is $\det \left(\mathit{I}\right)=1$

• The determinant of a zero matrix is $\det \left(\mathit{O}\right)=0$

• When finding the determinant of a multiple of a matrix or the product of two matrices:

  • $\det \left(k\mathit{A}\right)=k^n\det \left(\mathit{A}\right)$(for a $n\times n$ matrix)

  • $\det \left(\mathit{A}\mathit{B}\right)=\det \left(\mathit{A}\right)\times \det \left(\mathit{B}\right)$

  • $\det \left({\mathit{A}}^{\mathbf{-}\mathbf{1}}\right)=\frac{1}{\det \left(\mathit{A}\right)}$

• If $\det \left(\mathit{A}\right)=0$ then A is singular

• If $\det \left(\mathit{A}\right)\ne 0$ then A is non-singular