Determinant of a Matrix: Definition, Calculation & Examples

The determinant of a matrix is a scalar value that is calculated using the elements of a square matrix. It is a scaling factor for the transformation of a matrix. The determinant of a matrix is used to solve a system of linear equations, perform calculus operations, and calculate the inverse of a matrix.

• The square matrices can be a 2x2 matrix, 3x3 matrix, 4x4 matrix, or nxn matrix where the number of columns and rows are equal. 
• Matrix is represented as det (A) or |A|.
• The determinant of a matrix is a function mapping each square matrix to a unique real or complex number.
• If A is the set of all square matrices and B is the set of all numbers, then the determinant function f (f: A → B) is defined as f(x) = y, where y denotes the determinant of matrix 'x'.

​Also Read: Applications of Determinants and Matrices

Key Terms: Determinants, Square Matrices, column, Inverse of a matrix, Matrix, Complex number, Matrices, Linear equations

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What are Determinants?

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The determinant of a matrix is a scalar value calculated using a square matrix. In the case of every square matrix, A = \([a_{ij}]\) of order n, either a real or a complex number is associated, known as the determinant of the square matrix A, in which a = (i, j)^th element of A.

If M is the set of square matrices, K is the set of real or complex numbers, f: M → K is defined by f (A) = k, where k ∈ K and A ∈ M, then f (A) is the determinant of A.

So, if \(A =\begin{bmatrix} a & b \\ c & d \end{bmatrix}\)

Then, the determinant of matrix A, denoted by |A| = \(\begin{vmatrix} a & b \\ c & d \end{vmatrix}\) = det A

The symbol of the determinant of a matrix is Δ. It is represented by two vertical lines, or det and the matrix name. eg. |A|, det(A), det A.

The video below explains this:

Matrices Detailed Video Explanation:

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Singular and Non-Singular Matrix

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If the value of the determinants corresponding to the matrix is zero, it is said to be a square matrix, which means, 

• If |A| = 0, then the matrix is Singular Matrix
• And if |A| \(\neq\)0, then the matrix is a non-singular matrix

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Calculating the Value of a Determinant

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For every square matrix [A]_nxn, there is a determinant to the matrix that represents a unique value.

Step 1: Write out the matrix with its four entries. For example, a 2x2 matrix A can be written as:

A = [a b]

      [c d]

Step 2: Multiply the entries in the main diagonal (the top-left to bottom-right diagonal) and subtract the product of the off-diagonal entries (the top-right to bottom-left diagonal). In other words, compute:

det(A) = ad - bc

Step 3: The resulting number is the determinant of the matrix.

For example, consider the matrix: \(A =\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}\)

To calculate its determinant, we use the formula above: det(A) = (2 * 4) - (1 * 3) = 8 - 3 = 5.

Determinant of 2x2 Matrix

If we take det (A) = \(A =\begin{bmatrix} a11 & a21 \\ a22 & a12 \end{bmatrix}\)is the matrix of 2x2 order, 

The above steps can be used to solve this matrix

So, det (A) = \(\bigtriangleup\) = a₁₁a₁₂ – a₂₁a₂₂

Determinant of 3x3 Matrix

In the case of calculating the value of a 3x3 matrix, let us take an example:

det (A) 

\(A = \begin{bmatrix}a & b & c\\[0.3em]d & e & f \\[0.3em]g & h & i\\[0.3em] \end{bmatrix}\)

Step 1: Find the three products of diagonal elements going from left to right:
a * e * i
b * f * g
c * d * h

Step 2: Find the three products of diagonal elements going from right to left:
c * e * g
b * d * i
a * f * h

Step 3: Subtract the sum of the products from step 3 from the sum of the products from step 2:
det = (a * e * i + b * f * g + c * d * h) - (c * e * g + b * d * i + a * f * h)

Step 4: Simplify and compute the result:
det = aei + bfg + cdh - ceg - bdi - afh

Step 5: The resulting value is the determinant of the 3x3 matrix.

The determinant is tried to be expanded along the row with the maximum number of zeroes in order to make the calculations easy.

Alternative Method

To find the determinant of a 3x3 matrix using the method of splitting into sub-matrices, follow these steps:

Step 1: Write down the matrix:
[a b c]
[d e f]
[g h i]

Step 2: Split the matrix into three 2x2 matrices, each of which excludes one row and one column of the original matrix:
A = [e f]    B = [d f]    C = [d e]
       [h i]          [g i]           [g h]

Step 3: Compute the determinants of each sub-matrix:
det(A) = (e * i) - (f * h)
det(B) = (d * i) - (f * g)
det(C) = (d * h) - (e * g)

Step 4: Use the formula to calculate the determinant of the 3x3 matrix using the determinants of the sub-matrices:

det = a * det(A) - b * det(B) + c * det(C)

Step 5: Note that the signs of the second and fourth terms alternate between addition and subtraction.

Step 6: Simplify and compute the result:

det = a(ei - fh) - b(di - fg) + c(dh - eg)

The resulting value is the determinant of the 3x3 matrix.

The diagram given below consists of an example of a solved 3x3 matrix.

Discover about the Chapter video:

Determinants Detailed Video Explanation:

Also Read: Types of Matrices

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Things to Remember

• In a square matrix, the number of rows is always equal to the number of columns.
• The 2x2 square matrix has two rows & two columns.
• For solving a 3x3 matrix, one row is needed to expand & then we can solve it as a 2x2 determinant.
• The determinant of a matrix always has a scalar value.
• The determinant of a matrix can be represented as |A| or det (A).
• Matrix is considered important in solving linear equations and useful for calculating the inverse of a matrix & calculus operations.
• There are usually 6 different ways of expanding a determinant of 3x3 order in which each of the three rows (R1, R2, R3) and three columns (C1, C2, C3) gives the same value. So, expanding any of one row or column will give out the same value.

Also Read:

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Previous Year Questions

1. Find the solution of the equation by using the determinant formula. [BITSAT 2006]
2. Evaluate the determinant of the two matrices A and t. [BITSAT 2006]
3. Apply determinant properties to solve the system of 3 equations. [AP EAPCET]
4. Find the constant term in the polynomial by applying property of determinant. [KCET 2010] 
5. If two matrices are in A.P and find the range of the determinant. [CBSE Clas XII]
6. A.adj(A) is equal to...[KCET 2007]
7. If A=[α2​2α​] | A^3|= 27∣A3∣=27then α is equal to….[KCET 2015]
8. If A is a 3×3 nonsingular matrix and if |A | = 3, then |∣(2A)−1∣= _______[KCET 2010]
9. If A is a matrix of order 3, such that A (adj A) = 10, then |adj A|=….​[KCET 2014]
10. If A is a square matrix of order 3 × 3, then ∣KA∣ is equal to….[KCET 2017]