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Maths > Determinants > Adjoint and Inverse of a Matrix
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 the role of Adjoint in finding the Inverse of a matrix and some of its theorems.

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Definition of Adjoint of a Matrix

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. Adjoing of the matrix A is denoted by adj A.

(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

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


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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 identitiy 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

Solved Examples for You

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 if not equal to 0. For option 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 id the determinant is 0 then it has no matrix. After that, rearrange the matrix by rewriting the first row as the first column, middle row as middle column and final row as the final column.

Question 3: Explain the Cramer’s rule matrices?

Answer: For a 2×2 (with two variable) Cramer’s Rule is another method than 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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jannat

Is determinant available just for square matrix?

Stephen
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Stephen

Yes, it is only defined for square matrices.

J. A. Zahálka
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J. A. Zahálka

It actually is. It is equal to zero for all non-square matrices.

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