# DETERMINANT OF 1X1 AND 2X2 MATRIX

The determinant is the scalar value that is a function of the entries of a square matrix.

The determinant of a matrix, say A, is usually denoted as Det(A) or |A|

We will be looking at the determinant of the 1x1 matrix and the 2x2 matrix.

## Determinant of a 1x1 matrix

The determinant of a 1x1 matrix is simply that number itself.

That is, if,

$A=\begin{bmatrix} 6 \end{bmatrix}$,

then $|A|=6$

Example 1

Given that $A=\begin{bmatrix} 9 \end{bmatrix}$ and $B=\begin{bmatrix} 6 \end{bmatrix}$, Solve $|A| +|B|$

Solution:

Remember that the determinant of a 1x1 matrix is the number itself, therefore,

$|A| +|B|=9+6=15$

## Determinants of a 2x2 matrix

The determinant of a 2x2 matrix, $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is defined as

$$\left(a(d)-b(c)\right)$$

In words, the determinant of a 2x2 matrix is the difference between the product of top left and bottom right entries, and the product of top right and bottom left entries.

Example 2

Find the determinant of $A=\begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$

Solution:

Recalled $\left(a(d)-b(c)\right)$, according

$|A|=\left(2(8)-4(6)\right)$

$|A|=16-24=-8$

Example 3

Find the determinant of $A=\begin{bmatrix} 9 & 9 \\ 2 & 7 \end{bmatrix}$

Solution:

$|A|=\left(9(7)-9(2)\right)$

$|A|=63-18=-45$

READ ALSO: MULTIPLICATION OF 2X2, 3X3 MATRICES

Example 4

Calculate the sum of det(A) and det(B) given the $A=\begin{bmatrix} 5 & 4 \\ 3 & 6 \end{bmatrix}$ and $B=\begin{bmatrix} 5 & 12 \\ 4 & 8 \end{bmatrix}$,

Solution:

Before we solve this, please note that det(A) is the same as |A| and det (B) is the same as |B|.

$det(A)=5(6)-4(3)$

$det(A)=30-12=18$

$det(B)=5(8)-12(4)$

$det(B)=40-48=-8$

$det(A)+det(B)=18-(-8)=26$

Example 5

Obtain the determinant of $B=\begin{bmatrix} \frac{5}{6} & \frac{4}{3} \\ \frac{2}{3} & \frac{3}{4} \end{bmatrix}$

Solution:

$\frac{5}{6}\left(\frac{3}{4}\right)-\frac{4}{3}\left(\frac{2}{3}\right)$

$\frac{15}{24}-\frac{8}{9}=\frac{-19}{72}$

To summarize, the determinant of a 1x1 matrix is that number itself whereas the determinant of a 2x2 matrix is ad-bc

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