Inverse Matrix

An inverse matrix of a square matrix A is a matrix $A^{-1}$ such that when multiplied with A, it results in the identity matrix I: $A.A^{-1}=A^{-1}.A=I$.

A matrix is invertible if its determinant is non-zero ($|A|\ne 0$). If the inverse exists, it is unique. Inverse matrices are crucial for solving systems of linear equations and performing various matrix operations.

1.0What is a Matrix?

A matrix is a rectangular array of numbers or elements arranged in rows and columns. It is typically represented as $\left(\begin{array}{cccc}{a}_{11}& a_{12}& ...& a_{1n}\\ a_{21}& a_{22}& ...& a_{2n}\\ ...& ...& ...& ...\\ a_{m1}& a_{m2}& ...& a_{1n}\end{array}\right)$, where m is the number of rows and n is the number of columns. Matrices are used in various fields such as mathematics, physics, computer science, and engineering, particularly for solving systems of linear equations, transformations, and data representation.

2.0Inverse of a Matrix

A square matrix A is said to be invertible (or non-singular) if and only if its determinant is non-zero, i.e., $|A|\ne 0$. In this case, there exists a matrix B such that: 

AB = I = BA

Here, B is called the inverse (or reciprocal) of A and is denoted as $A^{-1}$. Therefore, the relationship can be expressed as:

 $A^{-1}=B$ $\text{if and only if}AB=I=BA$

Additionally, the following identity holds for any invertible matrix A: $A.adj(A)=|A|I_n$

Multiplying both sides by , we get: $A^{-1}.A.(adjA)=A^{-1}.|A|I_n$

This simplifies to:

Thus, we have the necessary and sufficient condition for a square matrix A to be invertible: $|A|\ne 0$.

$\therefore A^{-1}=\frac{\text{adj}A}{|A|}$

Key Points to Remember

• The inverse of a matrix exists only if the matrix is square (same number of rows and columns).
• A matrix must be non-singular (its determinant should not be zero) to have an inverse.

3.0Inverse Formula For a 2 × 2 Matrix

For a 2 × 2 matrix:

$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$

The formula to find the inverse of matrix A is:

 $A^{-1}=\frac{1}{ad-bc}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$

Here, ad - bc is the determinant of the matrix. If the determinant is zero, the matrix does not have an inverse.

4.0Inverse Formula For a 3 × 3 Matrix

If A is a 3 × 3 matrix:

$A=\left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)$

The inverse of a 3 × 3 matrix is calculated using the matrix of minors, cofactors, and adjugates, which is a more complex process but essential for larger matrices.

5.0How to Calculate the Inverse of a Matrix

Step-by-Step Guide for a 2 × 2 Matrix

Let's say we have the matrix A:

$A=\left(\begin{array}{cc}2& 3\\ 1& 4\end{array}\right)$

1. Find the Determinant (det A):

det(A) = (2)(4) - (3)(1) 

= 8 - 3 = 5 

2. Find the Adjugate of A:

For a 2 × 2 matrix, swap the elements on the main diagonal and change the signs of the off-diagonal elements. 

3. Multiply by 1/det(A): 

$A^{-1}=\frac{1}{5}\left(\begin{array}{cc}4& -3\\ -1& 2\end{array}\right)$

So, the inverse matrix is:

$A^{-1}=\left(\begin{array}{cc}\frac{4}{5}& \frac{-3}{5}\\ \frac{-1}{5}& \frac{2}{5}\end{array}\right)$

Step-by-Step Guide for a 3 × 3 Matrix

Finding the inverse of a 3 × 3 matrix requires more steps, but it involves finding the determinant, the matrix of minors, and then using the cofactor and adjugate methods. You can refer to detailed examples or use online tools to calculate it faster.

Example of Inverse Matrix

Example 1: 2 × 2 Matrix

Given the matrix:$A=\left(\begin{array}{cc}5& 7\\ 2& 3\end{array}\right)$

Step 1: Calculate the determinant:

$det(A)=(5)(3)–(7)(2)$

$=15–14=1$

Step 2: Swap the elements of the diagonal and negate the off-diagonal elements:

$A^{-1}=\frac{1}{1}\left(\begin{array}{cc}3& -7\\ -2& 5\end{array}\right)$

$A^{-1}=\left(\begin{array}{cc}3& -7\\ -2& 5\end{array}\right)$

Example 2: 3 × 3 Matrix

For the matrix A =$\left[\begin{array}{ccc}4& -4& 5\\ -2& 3& -3\\ 3& -3& 4\end{array}\right]$. Find A^–1.

Solution:

$A=\left[\begin{array}{ccc}4& -4& 5\\ -2& 3& -3\\ 3& -3& 4\end{array}\right]$

Step 1: Calculate the Determinants

det (A) = 4 [3·4 – (–3)(–3)] – (–4) [(–2)(4) – (– 3)(3)] + 5 [(– 2) (–3)– 3. 3]

= 4 (12 – 9) – (– 4) (– 8 + 9) + 5 (+6 –9) 

= 4 (3) + 4 (1) + 5 (– 3)

= 12 + 4 – 15 

= 1

|A| = 1.

Minor of A 

$M_{11}=\left|\begin{array}{cc}3& -3\\ -3& 4\end{array}\right|=12-9=3.$

$M_{12}=\left|\begin{array}{cc}-2& -3\\ 3& 4\end{array}\right|=-8+9=1.$

$M_{13}=\left|\begin{array}{cc}-2& 3\\ 3& -3\end{array}\right|=6-9=-3.$

$M_{21}=\left|\begin{array}{cc}-4& 5\\ -3& 4\end{array}\right|=-16+15=-1.$

$M_{22}=\left|\begin{array}{cc}4& 5\\ 3& 4\end{array}\right|=16-15=1.$

$M_{23}=\left|\begin{array}{cc}4& -4\\ 3& -3\end{array}\right|=-12+12=0.$

$M_{31}=\left|\begin{array}{cc}-4& 5\\ 3& -3\end{array}\right|=12-15=-3.$

$M_{32}=\left|\begin{array}{cc}4& 5\\ -2& -3\end{array}\right|=-12+10=-2.$

$M_{33}=\left|\begin{array}{cc}4& -4\\ -2& 3\end{array}\right|=12-8=4.$

$[M_{ij}]=\left[\begin{array}{ccc}3& 1& -3\\ -1& 1& 0\\ -3& -2& 4\end{array}\right]$

Step 3: Cofactor of Matrices

$[C_{ij}]=\left[\begin{array}{ccc}3& -1& -3\\ 1& 1& 0\\ -3& 2& 4\end{array}\right]$

Step 4: Adjoint of A.

$\text{Adj. }A=[C_{ij}{]}^T=\left[\begin{array}{ccc}3& 1& -3\\ -1& 1& 2\\ -3& 0& 4\end{array}\right]$

Step 5. Applying formula.

$A^{-1}=\frac{\text{Adj}(A)}{|A|}$

$\Rightarrow A^{-1}=\frac{1}{1}\left[\begin{array}{ccc}3& 1& -3\\ -1& 1& 2\\ -3& 0& 4\end{array}\right]$

$\Rightarrow A^{-1}=\left[\begin{array}{ccc}3& 1& -3\\ -1& 1& 2\\ -3& 0& 4\end{array}\right]$

Example 3:

If $A=\left[\begin{array}{ccc}0& -1& 2\\ 2& -2& 0\end{array}\right],B=\left[\begin{array}{cc}0& 1\\ 1& 0\\ 1& 1\end{array}\right]$and M = AB, then M^-1 is equal to

$\text{(A)}\left[\begin{array}{cc}2& -2\\ 2& 1\end{array}\right]$

$\text{(B)}\left[\begin{array}{cc}\frac{1}{3}& \frac{1}{3}\\ -\frac{1}{3}& \frac{1}{6}\end{array}\right]$

$\text{(C)}\left[\begin{array}{cc}\frac{1}{3}& -\frac{1}{3}\\ \frac{1}{3}& \frac{1}{6}\end{array}\right]$

$\text{(D)}\left[\begin{array}{cc}\frac{1}{3}& -\frac{1}{3}\\ -\frac{1}{3}& \frac{1}{6}\end{array}\right]$

Ans. (C)

Solution:

$M=\left[\begin{array}{ccc}0& -1& 2\\ 2& -2& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\\ 1& 1\end{array}\right]=\left[\begin{array}{cc}1& 2\\ -2& 2\end{array}\right]$

$|M|=6,\text{adj }M=\left[\begin{array}{cc}2& -2\\ 2& 1\end{array}\right]$

$M^{-1}=\frac{1}{6}\left[\begin{array}{cc}2& -2\\ 2& 1\end{array}\right]=\left[\begin{array}{cc}\frac{1}{3}& -\frac{1}{3}\\ \frac{1}{3}& \frac{1}{6}\end{array}\right]$

6.0Properties Of The Inverse Of A Matrix

1. Product of Invertible Matrices: If A and B are invertible matrices of the same order, then the inverse of their product is given by: $(AB{)}^{-1}=B^{-1}{A}^{-1}$

Note: If $A_1,A_2,....,A_n$ are invertible square matrices of order n, then:

$(A_1{A}_2....A_n{)}^{-1}=A_n^{-1}{A}_{n-a}^{-1}....A_2^{-1}{A}_1^{-1}$

2. Transpose of an Invertible Matrix: If A is an invertible matrix, then its transpose $A^T$is also invertible, and: $(A^T{)}^{-1}=(A^{-1}{)}^T$
3. Inverse Properties:

(a) The inverse of the inverse of AA is AA, i.e., $(A^{-1}{)}^{-1}=A$.

(b) For any positive integer k, the inverse of $A^k$is , $k(A^{-1}{)}^k=A^{-k}$

4. Determinant of the Inverse: If A is an invertible matrix, then:$|A^{-1}|=\frac{1}{|A|}$
5. Idempotent Matrix: If an idempotent matrix (i.e., a matrix A such that $A^2=A$) is invertible, its inverse must be the identity matrix.
6. Nilpotent Matrix: A nilpotent matrix, which satisfies for some positive integer k, cannot be invertible because its determinant is zero.
7. Orthogonal Matrix: An orthogonal matrix A is always invertible, and its inverse is its transpose: $A^{-1}=A^T$
8. Involutory Matrix: For an involutory matrix A, which satisfies $A^2=I$, we have: $A=A^{-1}$

7.0Cancellation Law

Let A, B, C be square matrices of the same order n. If A is a non-singular matrix (i.e., $|A|\ne 0$), then the following cancellation laws hold:

• Left Cancellation Law: If AB = AC, then B = C.
• Right Cancellation Law: If BA = CA, then B = C.

Note: These cancellation laws hold only when A is non-singular, meaning .

8.0Sample Questions on Inverse Matrix 

1. How do you find the inverse of a matrix?

Ans: For a 2 × 2 matrix $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, the inverse is given by:

$A^{-1}=\frac{1}{|A|}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$

where |A| = ad – bc.

For larger matrices, you can use methods like Gaussian elimination, adjoint method, or matrix inversion formulas.

2. What is the adjoint of a matrix, and how is it related to the inverse?

Ans: The adjoint (or adjugate) of a matrix A is the transpose of the cofactor matrix of A. The inverse of A can be expressed as: $A^{-1}=\frac{1}{|A|}.adj(A)$ where |A| is the determinant of A.