Determinants provide valuable information about the properties of matrices and the systems they represent. The determinant of a 2×2 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 2×2 matrix, and use it to solve practice problems.
LINEAR ALGEBRA
Relevant for…
Learning about the determinant of 2×2 matrices with examples.
LINEAR ALGEBRA
Relevant for…
Learning about the determinant of 2×2 matrices with examples.
How to find the determinant of a 2×2 matrix?
To find the determinant of a 2×2 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 2×2 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 2×2 matrix A.
Solved exercises on determinant of 2×2 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 2×2 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 2×2 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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Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
