Determinants and matrices are used to solve linear equations by using Cramer’s rule or the Matrix method. You can compute determinants for square matrices only. They are also used to find the adjoint, and inverse of a matrix.
Let’s understand what is determinant of a matrix and how to calculate it using examples.
What is Determinant of a Matrix?
To every square matrix $\text{A} = [a_{ij}]$ of order $n$, you can associate a number (real or complex) called the determinant of the square matrix $\text{A}$, where $a_{ij} = (i, j)^{th}$ element of $\text{A}$.
This may be thought of as a function that associates each square matrix with a unique number (real or complex). If $\text{M}$ is the set of square matrices, $\text{K}$ is the set of numbers (real or complex), and $f : \text{M} \rightarrow \text{K}$ is defined by $f (\text{A}) = k$, where $\text{A} \in \text{M}$ and $k \in \text{K}$, then $f(\text{A})$ is called the determinant of $\text{A}$. It is also denoted by $| \text{A}|$ or $\text{det A}$ or $\triangle \text{A}$.
Note:
- For matrix $\text{A}$, $| \text{A}|$ is read as a determinant of $\text{A}$ and not a modulus of $\text{A}$
- Only square matrices have determinants
How to Find the Determinant of a Matrix?
For the simplest square matrix of order $1 \times 1$ matrix, which only has only one number, the determinant becomes the number itself. The determinants of higher-order matrices are calculated by splitting them into lower-order square matrices.
Determinant of 1×1 Matrix
If $\text{A} = [a]$ is a matrix of order $1$, then the determinant of $\text{A}$ is defined to be equal to $a$.
Examples of Determinant of 1×1 Matrix
Example 1: Find the determinant of a matrix $[-7]$.
Determinant of $[-7] = |-7| = -7$.
Note:
- Determinant of $[-a] = -a$
- Modulus of $-a = a$
Example 2: Find the determinant of a matrix $[9]$.
Determinant of $[9] = |9| = 9$.
Determinant of 2×2 Matrix
For any $2 \times 2$ square matrix or a square matrix of order $2 \times 2$, we can use the determinant formula to calculate its determinant. If $\text{A}$ is a $2 \times 2$ matrix, such that
then, its $2 \times 2$ determinant can be calculated as
Examples of Determinant of 2×2 Matrix
Example 1: Find the determinant of the given matrix.
Determinant of matrix $\text{A} = |\text{A}| = 5 \times 4 – 3 \times 2 = 20 – 6 = 14$.
Example 2: Find the determinant of the given matrix.
Determinant of matrix $\text{B} = |\text{B}| = -1 \times 9 – 7 \times 6 = -9 – 42 = -51$.
Example 3: Find the determinant of the given matrix.
Determinant of matrix $\text{C} = |\text{C}| = 4 \times 12 – (-3) \times 8 = 48 + 24 = 72$.
Determinant of 3×3 Matrix
For any $3 \times 3$ square matrix or a square matrix of order $3 \times 3$,
then, its $3 \times 3$ determinant can be calculated as $a_1 \times \left( b_2 c_3 – b_3 c_2 \right) – b_1 \times \left( a_2 c_3 – a_3 c_2 \right) + c_1 \times \left( a_2 b_3 – a_3 b_2 \right)$
Examples of Determinant of 3×3 Matrix
Example 1: Find the determinant of the given matrix.
Determinant of matrix $\text{A} = |\text{A}| = 2 \times (4 \times 5 – 2 \times 2) – 1 \times (1 \times 5 – 2 \times 3) + 3 \times (1 \times 2 – 4 \times 3)$
$= 2 \times (20 – 4) – 1 \times (5 – 6) + 3 \times (2 – 12)$
$= 2 \times 16 – 1 \times (-1) + 3 \times (-10)$
$= 32 + 1 – 30 = 3$
Example 2: Find the determinant of the given matrix.
Determinant of matrix $\text{B} = |\text{B}| = 3 \times (7 \times (-1) – 1 \times 8) – (-2) \times (2 \times (-1) – 1 \times 4) + 0 \times (2 \times 8 – 7 \times 4)$
$= 3 \times (-7 – 8) + 2 \times (-2 – 4) + 0$
$= 3 \times (-15) + 2 \times (-6) + 0$
$= -45 – 12 + 0 = -57$
Key Takeaways
- A determinant can be considered as a function that takes a square matrix as the input and returns a single number as its output.
- A square matrix can be defined as a matrix that has an equal number of rows and columns.
- For the simplest square matrix of order $1 \times 1$ matrix, which only has only one number, the determinant becomes the number itself.
Practice Problems
Find the determinant of the following $1 \times 1$ matrices: [-9], [0], [5]
Find the determinant of the following $2 \times 2$ matrices
Find the determinant of the following $3 \times 3$ matrices
FAQs
What is determinant?
To every square matrix $\text{A} = [a_{ij}]$ of order $n$, you can associate a number (real or complex) called the determinant of the square matrix $\text{A}$, where $a_{ij} = (i, j)^{th}$ element of $\text{A}$.
This may be thought of as a function that associates each square matrix with a unique number (real or complex). If $\text{M}$ is the set of square matrices, $\text{K}$ is the set of numbers (real or complex), and $f : \text{M} \rightarrow \text{K}$ is defined by $f (\text{A}) = k$, where $\text{A} \in \text{M}$ and $k \in \text{K}$, then $f(\text{A})$ is called the determinant of $\text{A}$. It is also denoted by $| \text{A}|$ or $\text{det A}$ or $\triangle $.
What are determinants used for?
Determinants play an important role in linear equations where they are used to capture variables change in integers and how linear transformations change volume or area. Determinants are especially useful in applications where inverses and adjoints of matrices are used. The cross-product of two vectors is also calculated using determinants.
What is the determinant formula for a 2×2 matrix?
For any $2 \times 2$ square matrix or a square matrix of order $2 \times 2$, we can use the determinant formula to calculate its determinant. If $\text{A}$ is a $2 \times 2$ matrix, then $|\text{A}| = ad – bc$.
Are determinants commutative?
Yes, the multiplication of determinants is commutative and this can be well understood with this property: If $\text{A}$ and $\text{B}$ are two square matrices with order $n \times n$, then $\text{det}(\text{AB}) = \text{det}(\text{A}) \times \text{det}(\text{B}) = \text{det}(\text{B}) \times \text{det}(\text{A})$.
Conclusion
The determinant of a square matrix, $\text{A} = [a_{ij}]$ of order $n \times n$, can be defined as a scalar value that is real or a complex number, where $a_{ij}$ is the $(i,j)^{th}$ element of matrix $\text{A}$. It is denoted as $\text{det}(\text{A})$ or $| \text{A}|$. Determinants play an important role in linear equations where they are used to capture variables change in integers and how linear transformations change volume or area. Determinants are especially useful in applications where inverses and adjoints of matrices are used. The cross-product of two vectors is also calculated using determinants.
