Determinants

Adjoint and Inverse of a Matrix

Did you know that the Inverse of a Matrix can be easily calculated using the Adjoint of a Matrix? Having said that I would also like to bring your attention to the fact that the Inverse of a Matrix exists if and only if the value of its determinant is equal to zero. To begin with let’s look into finding the Inverse of a matrix and some of its theorems.

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Inverse of Matrix

Just like a number has it’s reciprocal, even a matrix has an inverse. If we consider a matrix A, we denote its inverse as A-1. The inverse of a matrix is another matrix that, when multiplied by the given matrix, yields the multiplicative identity.

For a matrix A, its inverse is A-1. And A.A-1 = I, where I is denoted as the identity matrix. In order to find the inverse matrix, the square matrix must be non-singular and have a determinant value that is not zero. Let us consider a 2×2 square matrix A.

Matrix A

The determinant of matrix A is denoted as ad-bc, and the value of the determinant should not be zero in order for the inverse matrix of A to exist. A simple formula can be used to calculate the inverse of a 2×2 matrix. Furthermore, in order to obtain the inverse of a 3×3 matrix, we must first determine the determinant and adjoint of the matrix.

Identity Matrix

The Identity Matrix (I) is obtained by multiplying a matrix by its inverse. The Identity Matrix is a matrix with a value of one. An Identity Matrix is a diagonal matrix in which all diagonal components are equal to 1 and the rest are equal to 0. It is also called a Unit Matrix or an Elementary Matrix.

The identity matrix for the 2×2 matrix is given by,

Identity Matrix

Terms related to Inverse of Matrix

  • Minor

The determinant matrix created after eliminating the row and column of the matrix in which that particular element lies is defined as the minor of that element in the matrix. Minor of an element aij is denoted by Mij.

For example, consider a matrix A.

Minor 1

The minor of element a12 is,

Minor 2

Thus eliminating Row 1 and Column 2 from the original matrix to get the resultant minor.

  • Cofactor

A cofactor is a number derived by removing the row and column of a certain element in a matrix, which is a numerical grid in the shape of a rectangle or a square. A positive (+) or negative (-) sign usually precedes the cofactor. The formula for determining the cofactor for a particular element is as follows,

Aij = (-1)i+j det Mij

Let us find the cofactor for A31 from the below-given matrix A.

First, we need to find the minor of M31 by eliminating Row 3 and Column 1.

Then we use the formula for calculating the cofactor,

A31 = (-1)3+1 x (-3)

A31 = -3

Hence, the cofactor of A31 is -3.

  • Determinant

The matrix determinant is the product of the elements of any row or column and their respective co-factors. Matrix determinants are only specified for square matrices. The determinant of any square matrix A is denoted by det A (or) |A|. It is sometimes represented by the sign Δ.

Let us look at the determinant of a 3×3 matrix.

Determinant of Matrix C is,


determinants cheat sheet

Browse more Topics under Determinants

 

  • Singular matrix

A singular matrix is a square matrix with a determinant value equal to zero. We cannot find the inverse of a singular matrix. For a singular matrix, |A| = 0.

  • Non-singular Matrix

A non-singular matrix is a square matrix with a non-zero determinant. To find the inverse of a matrix, the non-singular matrix property must be satisfied. For a non-singular matrix, |A| ≠ 0.

  • Adjoint Matrix

The adjoint of a matrix is generated by obtaining the transpose of the matrix’s co-factor members. The adjoint of a matrix is one of the simplest methods used for calculating a matrix’s inverse. The adjoint of a square matrix A = [aij]n x n is defined as the transpose of the matrix [Aij]n x n, where Aij is the cofactor of the element aij. Adjoining of the matrix A is denoted by adj A.

inverse of a matrix

(Image Source: tutormath)

Example 1

Find the adjoint of the matrix:

Solution: We will first evaluate the cofactor of every element,

Therefore,

(source: cliffnotes)

The Relation between Adjoint and Inverse of a Matrix

To find the inverse of a matrix A, i.e A-1 we shall first define the adjoint of a matrix. Let A be an n x n matrix. The (i,j) cofactor of A is defined to be

Aij = (-1)ij det(Mij),

where Mij is the (i,j)th minor matrix obtained from A after removing the ith row and jth column. Let’s consider the n x n matrix A = (Aij) and define the n x n matrix Adj(A) = AT. The matrix Adj(A) is called the adjoint of matrix A. When A is invertible, then its inverse can be obtained by the formula given below.

The inverse is defined only for non-singular square matrices. The following relationship holds between a matrix and its inverse:

AA-1 = A-1A = I, where I is the identity matrix.

Example 2

Determine whether the matrix given below is invertible and if so, then find the invertible matrix using the above formula.

\( \begin{bmatrix} 1 & 5 & 2 \\ 0 & -1 & 2 \\ 0 & 0 & 1 \end{bmatrix} \)

Solution: Since A is an upper triangular matrix, the determinant of A is the product of its diagonal entries. This, we have det(A) = -1, which is a non-zero value and hence, A is invertible. To find the inverse using the formula, we will first determine the cofactors Aij of A. We have,

Then the adjoint matrix of A is

\( \begin{bmatrix} -1 & -5 & 12 \\ 0 & 1 & -2 \\ 0 & 0 & -1 \end{bmatrix} \)

Using the formula, we will obtain the inverse matrix as

\( A^{-1} = adj(A)/det(A) =  \begin{bmatrix} 1 & 5 & -12 \\ 0 & -1 & 2 \\ 0 & 0 & 1 \end{bmatrix} \)

Few Rules for Row and Column Operations of a Determinant

  • If the rows and columns are swapped, the determinant’s value remains constant.
  • If any two rows or columns are swapped, the sign of the determinant changes.
  • If any two rows or columns of a matrix are equal, the determinant has a value of zero.
  • If every element of a given row or column is multiplied by a constant, the determinant’s value is also multiplied by the constant.
  • If the elements in a row or column can be expressed as a sum of elements, the determinant may be expressed as a sum of determinants.
  • If the elements of one row or column are added or subtracted with the matching multiples of elements from another row or column, the determinant value remains constant.

Methods to Find Inverse of Matrix

The inverse of a matrix can be found by using 3 different techniques. By using any of these 3 methods, the result obtained would be the same.

Method 1: For 2×2 matrix

Using the below formula, we can easily calculate the inverse of a 2×2 matrix.

We swap the positions of a and d, put a negative sign in front of b and c, and divide the matrix by its determinant, i.e. ad-bc. To check if the inverse matrix found is the right answer, we can multiply the inverse matrix with the original matrix. If the multiplication produces an identity matrix, then our answer is correct.

Method 2: Elementary Row/Column Operations

This method is also called the Gauss-Jordan method. We convert the given matrix into an identity matrix to find the inverse of a matrix using the elementary transformation method. We begin with the augmented matrix [A | I] or A = I A, where I is the identity matrix with the same order as A. The row/column operations are then used to transform the left side A to I. The matrix is then transformed into [I | A-1] or I = A-1 A.

For elementary row operations we can do the following:

  • swap rows.
  • multiply or divide each element in a row by a constant value.
  • a row can be replaced by adding or removing a multiple of another row from it

Let us look at an example to understand this method.

We use the statement A = I A

Multiply Row 1 by ½, we get

We then subtract Row 2 by Row 1,

We multiply Row 2 by 5 and then subtract Row 1 with Row 2,

Finally, we multiply Row 2 by 2,

Therefore, we have found the Inverse of the given matrix A.

Method 3: Using the Adjoint matrix and the Determinant

For a matrix A, its inverse A-1 = (1/|A|) x Adj A

Note – We will first check if the Matrix is a non-singular and invertible matrix i.e. |A| ≠ 0

The inverse of a matrix can be computed by following the steps below:

Step 1: Determine the minors of all the elements of matrix A.

Step 2: Next we compute the cofactors of all elements and build the cofactor matrix by substituting the elements of A with their respective cofactors.

Step 3: Take the transpose of A’s cofactor matrix to find its adjoint (written as adj A).

Step 4: Multiply adj A by the determinant’s reciprocal.

Inverse of 2×3 Matrix

A matrix must be non-singular in order to have an inverse matrix. For it to be non-singular, the determinant of a matrix must be non-zero. And the determinant only exists for square matrices. This indicates that where m x n, the inverse of matrices of order m ≠ n will not exist. As a result, we are unable to compute the inverse of the 2×3 matrix.

Important Points on Inverse of a Matrix

The following points will help you comprehend the concept of matrix inverse more fully.

  • The inverse of a square matrix is unique.
  • If A and B are two same-order invertible matrices, then (AB)-1 = B-1 A-1
  • Only if the determinant of a square matrix A is non-zero, i.e. |A| ≠ 0, does the inverse exist.
  • When the elements of a row or column are multiplied by the elements of any other row or column, their sum is zero.
  • The product of the determinants of two matrices is the product of the determinants of the two individual matrices. |AB| = |A||B|

Theorems on Adjoint and Inverse of a Matrix

Theorem 1

If A be any given square matrix of order n, then A adj(A) = adj(A) A = |A|I, where I is the identity matrix of order n.

Proof: Let

Since the sum of the product of elements of a row (or a column) with corresponding cofactors is equal to |A| and otherwise zero, we have

Similarly, we can show that adj(A) A = |A| I

Hence, A adj(A) = adj(A) A

Theorem 2

If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.

Theorem 3

The determinant of the product matrices is equal to the product of their respective determinants, that is, |AB| = |A||B|, where A and B are square matrices of the same order.

Remark: We know that

Writing determinants of matrices on both sides, we have

i.e,

i.e, |adj(A)| |A| = |A|3

or |adj(A)| = |A|2

In general, if A is a quare matrix of order n, then |adj(A)| = |A|n-1.

Theorem 4

A square matrix A is invertible if and only if A is a non-singular matrix.

Proof: Let A be an invertible matrix of order n and I be the identity matrix of the same order. Then there exists a square matrix B of order n such that AB = BA = I. Now, AB = I.

So |A| |B| = |I| = 1 (since |I| = 1 and |AB| = |A| |B|). This gives |A| to be a non-zero value. Hence A is a non-singular matrix. Conversely, let A be a non-singular matrix, then |A| is non-zero. Now A adj(A) = adj(A) A = |A| I (Theorem 1). Or,

or AB = BA = I, where

Thus A is invertible and

FAQs on Inverse of a Matrix

Question 1: With 1, ωω2 as cube roots of units, inverse of which of the following matrices exist?

  1. \( \begin{bmatrix} 1 & ω \\ ω & ω^{2} \end{bmatrix} \)
  2. \( \begin{bmatrix} ω^{2} & 1 \\ 1 & ω \end{bmatrix} \)
  3. \( \begin{bmatrix} ω & ω^{2} \\ ω^{2} & 1 \end{bmatrix} \)
  4. None of these.

Answer: Option D. Inverse of a matrix exists if its determinant is not equal to 0. For options A, B and C, the determinants are equal to zero, hence the inverse does not exist for any of the given matrices.

Question 2: How to find the inverse of a 3×3 matrix?

Answer: For finding the inverse of a 3×3 matrix, first of all, calculate the determinant of the matrix and if the determinant is 0 then it has no matrix. After that, rearrange the matrix by rewriting the first row as the first column, the middle row as the middle column and the final row as the final column.

Question 3: Explain Cramer’s rule matrices?

Answer: For a 2×2 (with two variables) Cramer’s Rule is another method that can solve systems of linear equations using determinants. Moreover, a matrix is an array of numbers enclosed by square brackets.

Question 4: Is inverse and transpose the same?

Answer: Matrix has an inverse if and only if it is both square and non-degenerate. Also, the inverse is unique. Besides, the inverse of an orthogonal matrix is its transpose. Moreover, they are the only matrices whose inverse are the same as their transpositions.

Question 5: Define a matrix?

Answer: Transpose refers to a matrix of an operative that tosses a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as \(A^{T} or {A}’, A^{tr}, ^{t}\textrm{A}\).

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