Determinant of a Matrix

Published: 11 Mar 2023

Get started with the Determinant of a matrix with this beginner-friendly post, covering the basics of the Determinant of a Matrix, Singular Matrix, Non-Singular Matrix, Adjoint and Multiplicative Inverse of a Matrix.

Let’s learn how to calculate the Determinant of a Matrix.

Determinant of a Matrix

What is Determinant of a Matrix

The determinant of a matrix is the scalar value or number that can be calculated from a square matrix that can provide essential information about matrix properties and behavior.

The matrix’s determinant helps us determine whether the Inverse of a Matrix exists or not, which can help us find the Systems of Linear Equations.

Symbol

The determinant of a matrix is represented in the following ways.

  • Write the name of the matrix in two vertical lines. i.e., $ |A| $
  • Write the det with the matrix name. i.e. $ det \ (A) \ or \ det \ A $.

Determinant of a Square Matrix

How to find the Determinant of a Square Matrix.
The determinant is a fundamental property of square matrices and can only be calculated from the square matrix. Square matrices are those matrices that have an equal number of rows and columns. The determinant of a square matrix is a single scalar value. The square matrix could be $ 1 \times 1, 2 \times 2, 3 \times 3 $ and much more…

Calculate the Determinant of a Matrix

To find the determinant of a matrix, it shall be noted that the matrix should be a square matrix. Following are the methods to calculate the determinant of a matrix.

Determinant of 1×1 Matrix

Determinant of 1×1 Matrix
The determinant of 1×1 matrix is the number itself. Let $ A=[a] $ be the matrix of order 1×1 matrix, then the determinant of A is: $ |A|=a $
Example of 1×1 Matrix
Find the determinant of $ C=[4] $.
Solution:
The determinant of $ 1 \times 1 $ matrix is itself. Thus: $ |C|=4 $

Determinant of 2×2 Matrix

Determinant of 2×2 Matrix
Let $ A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] $ be the matrix of 2×2; the following steps are required to find the determinant of the square matrix 2×2.
Steps for Determinant of 2×2 Matrix
Step 1: Write in Standard form
Convert the matrix in standard form to find the determinant of the matrix by changing the square brackets to vertical lines with |A|.
$ |A|=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right| $
Step 2: The Determinant of A is followed by:
The determinant of A equal to $ a \times d-c \times d $.
$ |A|=a d-c b $
Example:
$ A=\left[\begin{array}{cc}5 & 6 \\ -4 & 1\end{array}\right] $
Solution:
$ A=\left[\begin{array}{cc}5 & 6 \\ -4 & 1\end{array}\right] $
$ |A|=\left|\begin{array}{ll}5 & 6 \\ -4 & 1\end{array}\right| $
$ |A|=(5)(1)-(-4)(6) $
$ |A|=5-(-24) $
$ |A|=5+24 $
$ |A|=29 $

Determinant of 3×3 Matrix

Determinant of 3×3 Matrix
Let $ A=\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right] $ be the matrix of 3×3; the following steps are required to find the determinant of the square matrix 3×3.
Steps for Determinant of 3×3 Matrix
Step 1: Write in Standard form
Convert the matrix in standard form to find the determinant of the matrix by changing the square brackets to vertical lines with |A|.
$ |A|=\left|\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right| $
Step 2: To find the $ |F| $, expand the determinant along any Row or Column. Here we expand the determinant by R1 .
$ |A|=a\left|\begin{array}{ll}e & f \\ h & i\end{array}\right|-b\left|\begin{array}{ll}d & f \\ g & i\end{array}\right|+c\left|\begin{array}{ll}d & e \\ g & h\end{array}\right| $
$ |A|=a(ei-fh)-b(di-fg)+c(dh-eg) $
Example:
$ F=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] $
$ |F|=\left|\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right| $
Expand by Row 1:
$ |F|=1\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|-0\left|\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right|+0\left|\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right| $
$ |F|=1 \{ (1)(1)-(0)(0) \}-0 \{ (0)(1)-(0)(0) \}+0 \{ (0)(0)-(0)(1) \} $
$ |F|=1(1-0)-0(0-0)+0(0-0) $
$ |F|=1(1-0)-0+0 $
$ |F|=1(1) $
$ |F|=1 $

Singular Matrix

What is Singular Matrix
When the determinant of a square matrix is zero is called Singular Matrix. A Singular matrix is also called NOT invertible.
Mathematically,
If $ |A|=0 $ then $ A $ is Singular Matrix.
$ D=\left[\begin{array}{cc}-3 & 6 \\ 2 & -4\end{array}\right] $
Example:
$ |D|=\left|\begin{array}{cc} -3 & 6 \\ 2 & -4 \end{array}\right| \\ $
$ |D|=12-12 \\ $
$ |D|=0 $
Thus D is a singular matrix

Non-Singular Matrix

What is Non-Singular Matrix?
When the determinant of a square matrix is zero is called Singular Matrix. A Singular matrix is also called NOT invertible.
Mathematically,
If $ |A| \neq 0 $ then $ \mathrm{A} $ is Non-Singular Matrix.
Example
$ C=\left[\begin{array}{cc}3 a & -2 b \\ 2 a & b\end{array}\right] $
Solution:
$ C=\left[\begin{array}{cc}3 a & -2 b \\ 2 a & b\end{array}\right] $
$ |C|=\left|\begin{array}{cc}3 a & -2 b \\ 2 a & b\end{array}\right| $
$ |C|=3 a b-(-4 a b) $
$ |C|=3 a b+4 a b $
$ |C|=7 a b \neq 0 $

Adjoint of Square Matrix

What is Adjoint of a Matrix
Let $ A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] $
As change the places of $ a \ and \ d $ with each other and change the signs of $ b \ and \ c $. So
$ {Adj} A=\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right] $
$ B=\left[\begin{array}{cc}-3 & -1 \\ 2 & 3\end{array}\right] $
$ {Adj} \mathrm{B}=\left[\begin{array}{cc}3 & 1 \\ -2 & -3\end{array}\right] $
$ D=\left[\begin{array}{cc}-3 & 6 \\ 2 & -4\end{array}\right] $
$ {Adj} {D}=\left[\begin{array}{rr}-4 & -6 \\ -2 & -3\end{array}\right] $

MCQs

1. |A| is called as
O Determinant
O A symmetric
O Adioint
O None
Show Answer

Determinant
Explanation:
This is the way to write Determinant


2. $ |A|=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|= $
O $ a b+c d $
O $ a b+a d $
O $ abcd $
O $ a d-b c $
Show Answer

$ ad-bc $
Explanation:
In this way, find the Determinant


3. $ \left|\begin{array}{cc}-2 & 2 \\ 3 & 5\end{array}\right|= $
O 16
O $ -6 $
O $ -16 $
O None of these
Show Answer

$ -16 $
Explanation:
$ (-2)(5)-(3)(2) $
$ -10-6 $
$ -16 $


4. $ \left|\begin{array}{cc}4 & -2 \\ -2 & 1\end{array}\right|= $
O 16
O 0
O 2
O 4
Show Answer

0
Explanation:
$ (4)(1)-(-2)(-2) $
$ 4-4 $
$ 0 $


5. The determinant is a
O number
O Transpose
O Matrix
O Adjoint
Show Answer

number
Explanation:


6. $ \left|\begin{array}{cc}4 & -2 \\ -2 & 1\end{array}\right|= $
O Singular
O Non-singular
O Adioint
O None
Show Answer

Singular
Explanation:
$ (4)(1)-(-2)(-2) $
$ 4-4 $
$ 0 $
As Determinant is 0. Thus, it is singular matrix.


7. $ |A|=0 $
O Singular
O Non-singular
O Adioint
O None
Show Answer

Singular
Explanation:
When the Determinant of matrix is zero, then the matrix is singular matrix.


8. $ |A| \neq 0 $
O Singular
O Non-singular
O Adioint
O None
Show Answer

Non-singular
Explanation:
When the Determinant of matrix is not equal zero, then the matrix is Non-singular matrix.


9. If $ A=\left[\begin{array}{ll}7 & 8 \\ 3 & 2\end{array}\right] \ then \ adj \ A= $
O $ \left[\begin{array}{cc}2 & 8 \\ -3 & 7\end{array}\right] $
O $ \left[\begin{array}{cc}2 & -8 \\ -3 & 7\end{array}\right] $
O $ \left[\begin{array}{cc}7 & 8 \\ -3 & 2\end{array}\right] $
O None
Show Answer

$ \left[\begin{array}{cc}2 & -8 \\ -3 & 7\end{array}\right] $
Explanation:
In Adjoint, change the places of a and d with each other and change the signs of b and c.


10. $ A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \ the \ adjA \ = $
O $ \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right] $
O $ \left[\begin{array}{ll}a & c \\ d & a\end{array}\right] $
O $ a d-b c $
O All of these
Show Answer

$ \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right] $
Explanation:
In Adjoint, change the places of a and d with each other and change the signs of b and c.


11. Multiplicative inverse of $ A $ is
O AB
O B
O A
O $ A^{-1} $
Show Answer

$ A^{-1} $
Explanation:


12. $ A \cdot A^{-1}=A^{-1} \cdot A $
O I
O A
O A^{-1}
O None
Show Answer

$ I $
Explanation:
A matrix and its multiplicative inverse is equal to Multiplicative Identity “I”.


13. $ A^{-1}=\frac{1}{|A|} {adj} $ of
O B
O I
O A
O None
Show Answer

A
Explanation:
Definition of Multiplicative inverse


14. $ F^{-1}=\frac{1}{|F|} $ adj of
O A
O I
O F
O None
Show Answer

F
Explanation:
Definition of Multiplicative inverse


15. If $ A=\left[\begin{array}{cc}1 & 3 \\ 2 & -2\end{array}\right] \ then \ A^{-1}= $
O $ -\frac{1}{8}\left[\begin{array}{cc}-1 & -3 \\ 2 & 2 \end{array}\right] $
O $ -\frac{1}{8}\left[\begin{array}{cc} -2 & -3 \\ -2 & 1 \end{array}\right] $
O $ \frac{1}{8}\left[\begin{array}{cc} -2 & -3 \\ -2 & 1 \end{array}\right] $
O None of these
Show Answer


Explanation:
$ A=\left[\begin{array}{cc}1 & 3 \\ 2 & -2\end{array}\right] $
We have
$ A^{-1}=\frac{1}{|A|} \ adj of A $
$ |A|=\left|\begin{array}{ll}1 & 3 \\ 2 & -2\end{array}\right| $
$ |A|=(1)(-2)-(2)(-3) $
$ |A|=-2-6 $
$ |A|=-8 $
$ Adj \ A=\left[\begin{array}{cc}-2 & -3 \\ -2 & 1\end{array}\right] $
Now
$ A^{-1}=\frac{1}{-8} \left[\begin{array}{cc}-2 & -3 \\ -2 & 1\end{array}\right] $

Review Exercise # 1 MCQS

1. $ \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] $ is
a. An identity matrix w.r.t multiplication
b. A column matrix
c. An identity matrix w.r.t addition
d. A row matrix
Show Answer

An identity matrix w.r.t addition
Explanation:
Zero (0) is called additive identity. Thus Zero or Null matrix is additive identity matrix.


2. The matrix $ \left[\begin{array}{cc}4 & 0 \\ 0 & -12\end{array}\right] $ is
a. A scalar matrix
b. $ \quad 2 \times 3 $ matrix
c. A diagonal matrix
d. None of these
Show Answer

A diagonal matrix
Explanation:
A square matrix on which all elements are zero except diagonal elements is known as diagonal matrix.


3. If $ A=\left[\begin{array}{cc}-1 & -2 \\ 3 & 1\end{array}\right] $ , then adj A is equal to
a. $ \quad\left[\begin{array}{cc}-1 & -2 \\ 3 & 1\end{array}\right] $
b. $ \quad\left[\begin{array}{cc}1 & 2 \\ -3 & -1\end{array}\right] $
c. $ \left[\begin{array}{cc}-1 & 2 \\ 3 & 1\end{array}\right] $
d. $ \quad\left[\begin{array}{cc}1 & -2 \\ 3 & 1\end{array}\right] $
Show Answer

$ \quad\left[\begin{array}{cc}-1 & -2 \\ 3 & 1\end{array}\right] $
Explanation:
Let $ A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] $
As change the places of $ a \ and \ d $ with each other and change the signs of $ b \ and \ c $. So
$ Adj \ A=\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right] $


4. If A$ =\left[\begin{array}{ll}2 & 3 \\ 3 & 4\end{array}\right] $ then $ A^{-1}= $
a. $ \quad\left[\begin{array}{cc}4 & 3 \\ -3 & 2\end{array}\right] $
b. $ \quad\left[\begin{array}{cc}4 & -3 \\ -3 & 2\end{array}\right] $
c. $ \quad\left[\begin{array}{cc}-2 & 3 \\ 3 & -4\end{array}\right] $
d. $ \quad\left[\begin{array}{cc}-4 & 3 \\ 3 & -2\end{array}\right] $
Show Answer


Explanation:
$ A=\left[\begin{array}{ll} 2 & 3 \\ 3 & 4 \end{array}\right] $
$ A^{-1}=\frac{1}{|A|} { Adj } A $
$ |A|=\left|\begin{array}{ll}2 & 3 \\ 3 & 4\end{array}\right| $
|A|=8-9
$ |A|=-1 \neq 0 $
$ {Adj} A=\left[\begin{array}{cc}4 & -3 \\ -3 & 2\end{array}\right] $
Put the values in equation
$ A^{-1}=\frac{1}{-1}\left[\begin{array}{cc} 4 & -3 \\ -3 & 2 \end{array}\right] $
$ A^{-1}=-\left[\begin{array}{cc} 4 & -3 \\ -3 & 2 \end{array}\right] $
$ A^{-1}=\left[\begin{array}{cc} -4 & 3 \\ 3 & -2 \end{array}\right] $


5. For what value of d is the $ 2 \times 2 $ matrix $ \left[\begin{array}{cc}1 & 1.5 \\ 2 & d\end{array}\right] $ not invertible?
a. $ -0.6 $
b. 0
c. 0.6
d. 3
Show Answer

6
Explanation:
Singular matrix is also called NOT invertible.
Thus $ |A|=0 $
$ \left|\begin{array}{ll}5 & 1.5 \\ 2 & d\end{array}\right|=0$
$ 5 \times d-2 \times 1.5=0 $
$ 5d-3=0 $
$ 5d=3 $
$ d=\frac{3}{5} $
$ d=0.6 $


6. Suppose A and B are $ 2 \times 5 $ matrices, which of the following are the dimensions of the matrix $ A+B $ ?
a. $ 2 \times 5 $
b. $ 10 \times 10 $
c. $ \quad 7 \times 7 $
d. $ \quad 7 \times 1 $
Show Answer

$ 2 \times 5 $
Explanation:
For Addition of two matrices, the dimensions of the matrices must be same. Thus A+B have the dimensions $ 2 \times 5 $


7.Which of the following is the multiplicative inverse of $ \left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right] $ is
a. $ \quad\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right] $
b. $ \quad\left[\begin{array}{cc}1 & -2 \\ 0 & 1\end{array}\right] $
c. $ \quad\left[\begin{array}{cc}-1 & 2 \\ 0 & 1\end{array}\right] $
d. $ \quad\left[\begin{array}{ll}1 & -2 \\ 0 & -1\end{array}\right] $

Show Answer

$ \quad\left[\begin{array}{cc}1 & -2 \\ 0 & 1\end{array}\right] $
Explanation:
$Let \ A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right] $
$ A^{-1}=\frac{1}{|A|} { Adj } A $
$Let \ |A|=\left|\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right| $
|A|=1-0
$ |A|=1 \neq 0 $
$ {Adj} A=\left[\begin{array}{cc}1 & -2 \\ 0 & 1\end{array}\right] $
Put the values in equation
$ A^{-1}=\frac{1}{1}\left[\begin{array}{cc} 1 & -2 \\ 0 & 1 \end{array}\right] $
$ A^{-1}=\left[\begin{array}{cc} 1 & -2 \\ 0 & 1 \end{array}\right] $


8. The determinant of $ \left[\begin{array}{cc}4 & -1 \\ -9 & 2\end{array}\right] $ is
a. 17
b. 1
c. $ -1 $
d. $ \quad-17 $
Show Answer

$ |A|=-1 $
Explanation:
$Let \ |A|=\left|\begin{array}{ll}4 & -1 \\ -9 & 2\end{array}\right| $
|A|=(4)(2)-(-9)(-1)
|A|=8-9
$ |A|=-1 $


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jawad khalil
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