Calculator of Inverse of a 2x2 Matrix

Example №1. Finding the Inverse of a 2x2 Matrix

This solution was made using the calculator presented on the site.

Example №2. Finding the inverse of a 3x3 matrix

It is necessary to calculate a matrix A^-1, inverse to the given one:

A =		1	2
		1	3

Formula for calculating the inverse matrix:

A^-1 = 1 / det A *		A₁₁	A₂₁
		A₁₂	A₂₂

A₁₁ ... A₂₂ are numbers (algebraic additions) that will be calculated later.

It is impossible to divide by zero. Therefore, if the determinant of A is zero, then it is impossible to calculate inverse matrix.

Let's calculate the determinant A.

det A =		1	2
		1	3

= 1 * 3 - 2 * 1 = 3 - 2 = 1

Determinant A is not zero. It is possible to calculate inverse matrix.

Let's calculate numbers (algebraic additions) A₁₁ ... A₂₂

	1	2
	1	3

Row number 1
Column number 1

Row 1 and column 1
have been deleted

A₁₁

( -1) ^{1 + 1}

= 3

	1	2
	1	3

Row number 1
Column number 2

Row 1 and column 2
have been deleted

A₁₂

( -1) ^{1 + 2}

= -1

	1	2
	1	3

Row number 2
Column number 1

Row 2 and column 1
have been deleted

A₂₁

( -1) ^{2 + 1}

= -2

	1	2
	1	3

Row number 2
Column number 2

Row 2 and column 2
have been deleted

A₂₂

( -1) ^{2 + 2}

= 1

Result:

A^-1 = 1 / det A *		A₁₁	A₂₁
		A₁₂	A₂₂

A^-1 = 1 / 1 *		3	-2
		-1	1

A^-1 =		3	-2
		-1	1

Let's check the result

It is necessary to check that A^-1 * A = E.

	3	-2
	-1	1

	1	2
	1	3

	b₁₁	b₁₂
	b₂₁	b₂₂

b₁₁ = 3 * 1 + ( -2) * 1 = 3 - 2 = 1

	3	-2
	-1	1

	1	2
	1	3

	1	b₁₂
	b₂₁	b₂₂

b₁₂ = 3 * 2 + ( -2) * 3 = 6 - 6 = 0

	3	-2
	-1	1

	1	2
	1	3

	1	0
	b₂₁	b₂₂

b₂₁ = -1 * 1 + 1 * 1 = -1 + 1 = 0

	3	-2
	-1	1

	1	2
	1	3

	1	0
	0	b₂₂

b₂₂ = -1 * 2 + 1 * 3 = -2 + 3 = 1

	3	-2
	-1	1

	1	2
	1	3

	1	0
	0	1

= E

Thus, the found matrix A^-1 is inverse for the given matrix A.

Go to the solution of your problem

Service for Solving Linear Programming Problems

Example №1. Finding the Inverse of a 2x2 Matrix