Inverse of a Square Matrix

1.0What is the Inverse of a Square Matrix?

A square matrix has the same number of rows and columns (n × n). The Inverse of a square matrix A is a different matrix, which is written as $(A^{-1})$, and it is written as follows:

$A\times A^{-1}=A^{-1}\times A=I$

Where (I) is the same order as (A) and is the identity matrix. In the world of matrices, the Inverse matrix is like the reciprocal of numbers. It is essential for solving systems of linear equations, finding matrix equations, and other related problems .

2.0Conditions for the Existence of an Inverse

A square matrix (A) can be invertible (or is not singular) if and only if:

• The determinant of A is not zero: (|A| ≠ 0)
• A is a square matrix: Inverses can only be found in n × n matrices.

If (|A| = 0), the matrix is called singular and it does not have an inverse.

3.0Methods to Find Inverse of a Square Matrix

There are two main ways to find the inverse of a square matrix, especially for JEE-level problems:

Adjoint Method

This method is based on the formula: $A^{-1}=\frac{1}{|A|}\times \text{adj}(A)$

where (|A|) is the determinant and ($\text{adj}(A)$) is the adjugate (or adjoint) of (A).

Steps:

1. Find the determinant (|A|).
2. Find the matrix of cofactors.
3. To get ($\text{adj}(A)$), transpose the cofactor matrix.
4. Take the adjugate and divide it by the determinant.

Elementary Row Transformation Method

This is a practical method, especially for larger matrices or when solving systems of equations:

Steps:

1. Write the matrix (A) alongside the identity matrix of the same order, forming an augmented matrix ([A | I]).
2. Use elementary row operations (swap, multiply, add/subtract rows) to turn (A) into (I).
3. The matrix that replaces the identity matrix after these operations is ($A^{-1}$).

4.0How to Find Inverse of a 2×2 Matrix

Let the given 2×2 matrix be:

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

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

Here, (ad - bc) is the determinant of (A). If (ad - bc = 0), (A) has no inverse.

5.0How to Find Inverse of 3x3 Matrix

Finding the inverse of a 3×3 matrix is a common JEE-level question.

Suppose $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$

Steps:

1. Find the determinant det(A).
2. Compute all cofactors of each element.
3. Form the cofactor matrix.
4. Take its transpose to get the adjoint.
5. Apply the formula: $A^{-1}=\frac{1}{det(A)}.adj(A)$

Example: Find the Inverse of a 3×3 Matrix

$A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 4\\ 5& 6& 0\end{array}\right]$

Step 1: Find determinant

$det(A)$$$\begin{gathered} =1(1\cdot 0-4\cdot 6)-2(0\cdot 0-4\cdot 5)+3(0\cdot 6-1\cdot 5) \\ =1(0-24)-2(0-20)+3(0-5) \\ =-24+40-15=1 \end{gathered}$$

Since $det(A)=1\ne 0$ the inverse exists.

Step 2: Find cofactors and adjoint

$C_{11}=(1\cdot 0-4\cdot 6)=-24$

$C_{12}=-(0\cdot 0-4\cdot 5)=20$

$C_{13}=(0\cdot 6-1\cdot 5)=-5$

Similarly compute all cofactors, form the cofactor matrix, and transpose it to get adj(A).

Step 3: Apply formula

$A^{-1}=\frac{1}{\det (A)}\mathrm{adj}(A)=\frac{1}{1}\mathrm{adj}(A)=\mathrm{adj}(A)$

Thus, the inverse can be directly obtained since det⁡(A)=1.

6.0Solved Examples on Inverse of a Matrix

Example 1: Inverse of a 2×2 Matrix

Find the inverse of $A=\left[\begin{array}{cc}3& 4\\ 2& 5\end{array}\right]$

• $(|A|=3\times 5-4\times 2=15-8=7)$
• Adjoint: $\left[\begin{array}{cc}5& -4\\ -2& 3\end{array}\right]$
• Inverse: $A^{-1}=\frac{1}{7}\left[\begin{array}{cc}5& -4\\ -2& 3\end{array}\right]$

Example 2: Inverse of a 3×3 Matrix

Find the inverse of $B=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 4\\ 5& 6& 0\end{array}\right]$

• |B| = $1(1\times 0-4\times 6)-2(0\times 0-4\times 5)+3(0\times 6-1\times 5)$
• |B| = -24 + 40 - 15 = 1
• Cofactor matrix: $\left[\begin{array}{ccc}-24& 20& -5\\ 18& -15& 4\\ 5& -4& 1\end{array}\right]$
• Adjugate: $\left[\begin{array}{ccccccc}-24& 18& 520& -15& -4-5& 4& 1\end{array}\right]$
• Inverse: $B^{-1}=\left[\begin{array}{ccc}-24& 18& 5\\ 20& -15& -4\\ -5& 4& 1\end{array}\right]$

Example 3: Inverse by Elementary Row Transformation

Find the inverse of $C=\left[\begin{array}{cc}2& 1\\ 5& 3\end{array}\right]$

• Augment with identity: $\left[\begin{array}{cccc}2& 1& 1& 0\\ 5& 3& 0& 1\end{array}\right]$
• Row operations yield: $\left[\begin{array}{cc}3& -1\\ -5& 2\end{array}\right]$
• Inverse: $C^{-1}=\left[\begin{array}{cc}3& -1\\ -5& 2\end{array}\right]$

Example 4: Inverse of a 3×3 Matrix with Zero Elements

Find the inverse of $D=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 0\\ 2& 0& 1\end{array}\right]$

• $|D|=1(1\times 1-0\times 0)-0+2(0\times 0-1\times 2)=1-4=-3$
• Cofactor matrix: $\left[\begin{array}{ccc}1& 0& -2\\ 0& -3& 0\\ -2& 0& 1\end{array}\right]$
• Adjugate: $\left[\begin{array}{ccc}1& 0& -2\\ 0& -3& 0\\ -2& 0& 1\end{array}\right]$
• Inverse: $D^{-1}=\frac{1}{-3}\left[\begin{array}{ccc}1& 0& -2\\ 0& -3& 0\\ -2& 0& 1\end{array}\right]$

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