Inverse Matrix: Definition, Formulas, Method & Examples

The inverse of a matrix is a different matrix that, when multiplied by the original matrix, yields the multiplicative identity. A matrix is a function that consists of an ordered or structured rectangular array of numbers. The array values are referred to as matrix elements. The horizontal arrays in a matrix are known as rows, and the vertical arrays are known as columns. If the number of rows and columns in a matrix is a and b, respectively, the matrix's order is an x b, where a and b are the counting numbers. If A is a matrix of an x b order, then A^-1 represents the inverse of matrix A. It can only be discovered the matrix inverse for square matrices with equal numbers of rows and columns, such as 2 × 2, 3 × 3, and so on. In simple terms, the inverse matrix is obtained by dividing the adjugate of the supplied matrix by its determinant. 

Read more: Applications of determinants

KeyTerms: Inverse matrix, Square matrix, Row operations, Column operations, Elements, Determinants, Matrix, Counting Numbers, Array, Original matrix

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What is an Inverse Matrix?

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If A is a non-singular square matrix, then A^-1 is an inverse matrix that meets the following condition:

 AA^-1 = A^-1A = I, Where I is the Identity matrix

The video below explains this:

Matrices Detailed Video Explanation:

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Rank of the Matrix 

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The rank of the matrix is the maximum number of linearly self-determining column vectors in the matrix.

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Inverse Matrix Formula

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This can be represented mathematically as A^-1A = I, where A is a square matrix.

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Methods to Find Inverse of Matrix

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In general, the inverse of a matrix is obtained using two elementary matrix operations methods. Row and column transformations are used to perform the basic operations on a matrix.

Elementary Row Operations

Take three square matrices, X, A, and B, and use elementary row operations to determine the inverse of a matrix. The matrix equation is, A X = B. Also, X = A^-1 B is the matrix equation. Write down the matrix A as A = I A. Perform the basic row operations on both sides to obtain the identity matrix. After transformations, the final matrix in R.H.S.with "A" being the given matrix's inverse.

Elementary Column Operations

Take three square matrices, X, A, and B, and use elementary column operations to determine the inverse of a matrix. Here the matrix equation is, A X = B. Hence X = A^-1 B is the matrix equation. Carry out the basic row procedures. A = I A is one way to express the given matrix A. Perform the basic column operations on both sides to obtain the identity matrix. After transformations, “A” is the inverse of the given matrix of the final matrix.

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How to Measure the Inverse?

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In the following steps, one may estimate the inverse of the matrix:

• First, determine whether or not the determinant is greater than zero. After that, compute a minor matrix.
• Second, turn into a cofactor matrix.
• Third, obtain the matrix's adjugate or adjoint by taking the transpose of the cofactor matrix.
• Fourth: Finally, multiply by 1 / Determinant.

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Inverse Matrix-Related Terms

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• Cofactor: The cofactor of an element in order representation is computed by multiplying the minor with -1 to the exponent of the sum of the row and column elements.
• Matrix determinant: The determinant of a matrix is its solitary unique value representation. The determinant of the matrix can be calculated using any row or column of the supplied matrix. The determinant of the matrix is the product of the elements and their cofactors in a given row or column of the matrix.
• Singular Matrix: A singular matrix has a determinant of zero. For a single matrix A, |A| equals 0.
• Non-Singular Matrix: A non-singular matrix has a determinant value that is not equal to zero. For a non-singular matrix, |A| 0 A non-singular matrix is called an invertible matrix since its inverse can be computed.
• Matrix Adjoint: The adjoint of a matrix is the transpose of the cofactor element matrix of the supplied matrix.

Example:

\(\begin{bmatrix}6 & 2 & 3 \\[0.3em]3 & 1 &1 \\[0.3em]10 & 3&4 \\[0.3em] \end{bmatrix}\)

Find the inverse of the following matrix

Solution:

2 \(\begin{bmatrix}1 & 1 \\[0.3em]-1 & 2 \\[0.3em] \end{bmatrix}\) – 1 \(\begin{bmatrix}1 & 1 \\[0.3em]1 & 2 \\[0.3em] \end{bmatrix}\) + 1 \(\begin{bmatrix}1 & 1 \\[0.3em]1 & -1 \\[0.3em] \end{bmatrix}\)

|A| = 2(2 + 1) – 1(1) + 1(-2)

= 2(3) – 1 – 2

= 6 – 3

= 3

As A is a non-singular matrix A^-1 exists

|A| = 6(1) – 2(2) + 3(-1)

= 6 – 4 – 3

= 6 – 7

= – 1

|A| = – 10

Minor and cofactor of row 1

Minor of 6

\(\begin{bmatrix}1 & 1 \\[0.3em]3& 4 \\[0.3em] \end{bmatrix}\)

= (4 – 3)

= (1)

= 1

Cofactor of 6= +(1)= 1

Minor of 2

\(\begin{bmatrix}3 & 1 \\[0.3em]10 & 4 \\[0.3em] \end{bmatrix}\)

= (12 – 10)

= (2)

= 2

Cofactor of 2= (-2)= – 2

Minor of 3

\(\begin{bmatrix}3 & 1 \\[0.3em]10 & 3 \\[0.3em] \end{bmatrix}\)

= (9 – 10)

= ( – 1)

= – 1

Cofactor of 3 = + (-1) = – 1

Similarly

The minor of cofactor of row 2 are as follows

Minor of 3= -1

Cofactor of 3 = -(-1)= 1

Minor of 1= -6

Cofactor of 1 = +(-6)= -6

Minor of 1 = -2

Cofactor of 1 = -(-2)= 2

Minor and cofactor of row 3

Minor of 10 = -1

Cofactor of 10 = +(-1) = -1

Minor of 3 = -3

Cofactor of 3 = -(-3)= 3

Minor of 4 = 0

Cofactor of 4 = +(0) = 0

Cofactor Matrix = \(\begin{bmatrix}1 & -2 &1 \\[0.3em]-1 & -6 & 2 \\[0.3em]-1 & 3 & 0 \\[0.3em] \end{bmatrix}\)

Adjoint Matrix = \(\begin{bmatrix}1 & 1 &-1 \\[0.3em]-2 & -6 & 3 \\[0.3em]-1 & 2 & 0 \\[0.3em] \end{bmatrix}\)

A ^– 1 = 1/1 = \(\begin{bmatrix}-1 & -1 &1 \\[0.3em]2 & 6 & -3 \\[0.3em]1 & -2 & 0 \\[0.3em] \end{bmatrix}\)

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Things to Remember

• An inverse matrix is a matrix that, when multiplied by a given matrix, produces the identity matrix.
• This can be represented mathematically as A^-1A = I, where A is a square matrix.
• The equation for an inverse matrix is A(-1) = 1/(|A|)Adj(A)'.
• The inverse matrix does not exist if the determinant |A| equals 0.
• Only non-singular matrices have inverse matrices.
• The inverse matrix's determinant is equal to the reciprocal of the original matrix's determinant.

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