Determinants provide valuable information about the properties of matrices and the systems they represent. The determinant of a 2x2 matrix $latex A=\begin{pmatrix} a & b \\ c & d \end{pmatrix} $ is equal to $latex ∣A∣=(a\times d)−(b\times c )$.
In this article, we will explore the meaning of the determinant, delve into the step-by-step process of calculating the determinant for a 2x2 matrix, and use it to solve practice problems.
LINEAR ALGEBRA
Relevant for...
Learning about the determinant of 2x2 matrices with examples.
LINEAR ALGEBRA
Relevant for...
Learning about the determinant of 2x2 matrices with examples.
How to find the determinant of a 2x2 matrix?
To find the determinant of a 2x2 matrix, we have to multiply the element $latex b$ by $latex c$ and subtract from the product of $latex a$ and $latex d$.
Given a 2x2 matrix
$$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$
The determinant of A, denoted as $latex |A|$ or $latex \det(A)$, is calculated as follows:
Step 1: Multiply the elements of the main diagonal (from top left to bottom right):
$latex a \cdot d $
Step 2: Multiply the elements of the second diagonal (from top to bottom):
$latex b \cdot c $
Step 3: Subtract the product of step 2 (second diagonal) from the product of step 1 (main diagonal):
$$ |A| = (a \cdot d) - (b \cdot c) $$
The result of this calculation is the determinant of the 2x2 matrix A.
Solved exercises on determinant of 2x2 matrices
EXAMPLE 1
Find the determinant of the following matrix:
$$ A = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix} $$
Solution
Step 1: Identify the elements of the matrix. In this case, $latex a=3$, $latex b=4$, $latex c=5$ and $latex d=6$.
Step 2: Multiply the elements of the main diagonal (from top left to bottom right):
$$ a \cdot d = 3 \cdot 6 = 18 $$
Step 3: Multiply the elements of the second diagonal (from top to bottom):
$$ b \cdot c = 4 \cdot 5 = 20 $$
Step 4: Subtract the product of step 3 from the product of step 2:
$$ |A| = (a \cdot d) - (b \cdot c)$$
$$ = 18 - 20 = -2 $$
EXAMPLE 2
Find the determinant of the following matrix:
$$A=\begin{bmatrix}4 & 7\\-3 & 5\end{bmatrix}$$
Solution
As we have already seen how to find the determinant of a 2x2 matrix step by step from the previous example, we are going to solve this problem by simply applying the formula.
Using the formula $latex \det(A) = ad - bc$, where $latex a=4, b=7, c=-3$, and $latex d=5$, we have:
$$\det(A) = (4)(5) - (7)(-3) $$
$$= 20 + 21 = 41$$
EXAMPLE 3
Calculate the determinant of the following matrix:
$$B=\begin{bmatrix}-2 & 9\\4 & -6\end{bmatrix}$$
Solution
Again, we will just apply the determinant formula directly.
We apply the formula $latex \det(B) = ad - bc$, with the values $latex a=-2, b=9, c=4$, and $latex d=-6$:
$latex \det(B) = (-2)(-6) - (9)(4)$
$latex = 12 - 36 = -24$
EXAMPLE 4
Find the determinant of the following matrix:
$$C=\begin{bmatrix}3 & -5\\-1 & 2\end{bmatrix}$$
Solution
We can recognize the values $latex a=3, b=-5, c=-1$, and $latex d=2$. So, we use the formula $latex \det(C) = ad - bc$ with these values:
$$\det(C) = (3)(2) - (-5)(-1)$$
$$ = 6 - 5 = 1$$
EXAMPLE 5
What is the determinant of the following matrix?
$$A=\begin{bmatrix}5 & 8\\ 2 & 3\end{bmatrix}$$
Solution
We substitute the values $latex a=5, b=8, c=2$, and $latex d=3$ into the formula $latex \det(A) = ad - bc$:
$$\det(A) = (5)(3) - (8)(2) $$
$$= 15 - 16 = -1$$
EXAMPLE 6
Calculate the determinant of matrix B:
$$B=\begin{bmatrix}6 & -2\\ -4 & 3\end{bmatrix}$$
Solution
Using the formula $latex \det(B) = ad - bc$, with the values $latex a=6, b=-2, c=-4$, and $latex d=3$, we have:
$$ \det(B) = (6)(3) - (-2)(-4)$$
$$ = 18 - 8 = 10$$
EXAMPLE 7
What is the determinant of matrix C?
$$C=\begin{bmatrix}9 & 4\\ 7 & 3\end{bmatrix}$$
Solution
We apply the formula $latex \det(C) = ad - bc$, where $latex a=9, b=4, c=7$, and $latex d=3$:
$$ \det(C) = (9)(3) - (4)(7)$$
$$ = 27 - 28 = -1$$
EXAMPLE 8
Find the determinant of the following matrix:
$$A=\begin{bmatrix}\frac{1}{2} & -\frac{3}{4}\\ -\frac{1}{3} & \frac{5}{6}\end{bmatrix}$$
Solution
This is a more complicated problem because we have fractional values. However, we proceed in the same way:
Apply the formula $latex \det(A) = ad - bc$, with $latex a=\frac{1}{2}, b=-\frac{3}{4}, c=-\frac{1}{3}$, and $latex d=\frac{5}{6}$:
$$\det(A) = \left(\frac{1}{2}\right)\left(\frac{5}{6} \right) - \left(-\frac{3}{4}\right)\left(-\frac{1}{3}\right)$$
$$ = \frac{5}{12} - \frac{1}{4} = \frac{1}{6}$$
EXAMPLE 9
Calculate the determinant of the following matrix:
$$B=\begin{bmatrix}\sqrt{2} & \frac{3}{\sqrt{2}}\\ \sqrt{3} & \sqrt{6}\end{bmatrix}$$
Solution
In this case, we have square roots in the elements, but we use the same process.
Apply the formula $latex \det(B) = ad - bc$, with $latex a=\sqrt{2}, b=\frac{3}{\sqrt{2}}, c=\sqrt{3}$, and $latex d=\sqrt{6}$:
$$\det(B) = (\sqrt{2})(\sqrt{6}) - \left(\frac{3}{\sqrt{2}}\right)(\sqrt{3}) $$
$$= 2\sqrt{3} - \frac{3\sqrt{6}}{2} $$
$$= 2\sqrt{3} - 3\sqrt{\frac{3}{2}}$$
EXAMPLE 10
Find the determinant of the matrix:
$$C=\begin{bmatrix}2x & 3y\\ 4y & -5x\end{bmatrix}$$
Solution
In this case, we have variables in the elements, so the determinant will be an algebraic expression.
Using $latex a=2x, b=3y, c=4y$, and $latex d=-5x$ in the formula $latex \det(C) = ad - bc$:
$$\det(C) = (2x)(-5x) - (3y)(4y)$$
$$ = -10x^2 - 12y^2$$
Determinant of 2x2 matrices - Practice problems
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See also
Interested in learning more about arrays? You can look at these pages:
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