How to Find the Determinant of Matrix

Here you will learn what is determinant of matrix and formula for how to find the determinant of matrix of different order.

Let’s begin –

What is Determinant ?

If the equations \(a_1x + b_1\) = 0, \(a_2x + b_2\) = 0 are satisfied by the same value of x, then \(a_1b_2 – a_2b_1\) = 0. 

The expression \(a_1b_2 – a_2b_1\) is called a determinant of the second order, and it is denoted by

\(\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}\) 

A determinant of second order consists of two rows and two columns.

Next consider the system of equations \(a_1x + b_1y + c_1\) = 0, \(a_2x + b_2y + c_2\) = 0, \(a_3x + b_3y + c_3\) = 0

If these equations are satisfied by the same values of x and y, then on eliminating x and y we get,

\(a_1(b_2c_3 – b_3c_2)\) + \(b_1(c_2a_3 – c_3a_2)\) + \(c_1(a_2b_3 – a_3b_2)\) = 0

The expression on the left is called a determinant of the third order, and is denoted by

\(\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}\) 

How to Find the Determinant of Matrix

Determinant of Matrix of Order 1

If A = \([a_1]\) is a square matrix of order 1, then the determinant of A is defined as

| A | = \(a_1\)  or,  \(|a_1|\) = \(a_1\)

Determinant of Matrix of Order 2

If A = \(\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix}\) is a square matrix of order 2,

then the expression \(a_1b_2 – a_2b_1\) is defined as the determinant of A.

i.e. | A | = \(\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}\) = \(a_1b_2 – a_2b_1\)

Determinant of Matrix of Order 3

If A = \(\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\) is a square matrix of order 3,

then the expression \(a_1(b_2c_3 – b_3c_2)\) – \(b_1(a_2c_3 – a_3c_2)\) + \(c_1(a_2b_3 – a_3b_2)\) is defined as the determinant of A.

i.e. | A | = \(\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}\) 

= \(a_1\begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix}\) – \(b_1\begin{vmatrix} a_2 & c_2 \\ a_3 & c_3 \end{vmatrix}\) + \(c_1\begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \end{vmatrix}\)

= \(a_1(b_2c_3 – b_3c_2)\) – \(b_1(a_2c_3 – a_3c_2)\) + \(c_1(a_2b_3 – a_3b_2)\)


Related Questions

Find the determinant of \(\begin{vmatrix} sinx & cosx \\ -cosx & sinx \end{vmatrix}\).

Find the determinant of A = \(\begin{bmatrix} 3 & -2 & 4 \\ 1 & 2 & 1 \\ 0 & 1 & -1 \end{bmatrix}\).

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