Find Inverse of Matrix

To understand how to find the inverse of a matrix, it's essential to first grasp what a matrix is. A matrix is a structured rectangular array of numbers, known as elements. These elements are organized into horizontal rows and vertical columns. The dimensions of a matrix are denoted by its number of rows and columns. If a matrix has ‘a’ rows and ‘b’ columns, its order is expressed as a × b, where ‘a’ and ‘b’ are positive integers.

For a given matrix A of order, a × b, the inverse of matrix A, if it exists, is denoted as A⁻¹. The inverse of a matrix is a crucial concept in linear algebra and is defined such that when matrix A is multiplied by its inverse A⁻¹, the result is the identity matrix III, which is a square matrix with ones on the diagonal and zeros elsewhere. Mathematically, this is represented as:

A⋅ A⁻¹ = A⁻¹⋅A = I   

A matrix has an inverse only if it is a square matrix, meaning the number of rows is equal to the number of columns and its determinant is non-zero.

1.0Inverse of a Matrix Definition

If A is a non-singular (invertible) square matrix, then there exists an inverse matrix denoted by A^–1. This inverse matrix satisfies the condition:

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

where I is the identity matrix.

2.0How to Find the Inverse of a 3 × 3 Matrix:

To calculate the inverse of a matrix, follow these steps: 

1. Find the Matrix of Minors: Calculate the minor for each element of the original matrix.
2. Convert to the Matrix of Cofactors: Apply the checkerboard pattern of signs to the matrix of minors.
3. Find the Adjoint (Adjugate) of the Matrix: Take the transpose of the matrix of cofactors.
4. Multiply by $\frac{1}{\det (A)}$: Compute the determinant of the original matrix and multiply the adjoint matrix by the reciprocal of the determinant.

This process will yield the inverse of the original 3 × 3 matrix.

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

Solution: First of all we will calculate the matrix of Minors.

To find the inverse of a 3 × 3 matrix, let's go through the process in detail.

Step 1: Calculate the Matrix of Minors

Given a 3 × 3 matrix A:

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

The matrix of minors is obtained by calculating the minor for each element. The minor of an element is the determinant of the 2 × 2 submatrix formed by deleting the row and column of that element.

Minors of A:

1. Minor of a:

$M_{11}=\left|\begin{array}{ll}e& f\\ h& i\end{array}\right|=ei-fh$

2. Minor of b:

$M_{12}=\left|\begin{array}{ll}d& f\\ g& i\end{array}\right|=di-fg$

3. Minor of c:

$M_{13}=\left|\begin{array}{ll}d& e\\ g& h\end{array}\right|=dh-eg$

4. Minor of d:

$M_{21}=\left|\begin{array}{ll}b& c\\ h& i\end{array}\right|=bi-ch$

5. Minor of e:

$M_{22}=\left|\begin{array}{ll}a& c\\ g& i\end{array}\right|=ai-cg$

6. Minor of f:

$M_{23}=\left|\begin{array}{ll}a& b\\ g& h\end{array}\right|=ah-bg$

7. Minor of g:

$M_{31}=\left|\begin{array}{ll}b& c\\ e& f\end{array}\right|=bf-ce$

8. Minor of h:

$M_{32}=\left|\begin{array}{ll}a& c\\ d& f\end{array}\right|=af-cd$

9. Minor of i:

$M_{33}=\left|\begin{array}{ll}a& b\\ d& e\end{array}\right|=ae-bd$

Thus, the matrix of minors is:

$\mathrm{Minors}(A)=\left(\begin{array}{lll}ei-fh& di-fg& dh-eg\\ bi-ch& ai-cg& ah-bg\\ bf-ce& af-cd& ae-bd\end{array}\right)$

Step 2: Convert to the Matrix of Cofactors

Next, we apply the checkerboard pattern of signs to the matrix of minors to obtain the matrix of cofactors. The signs alternate starting with a positive sign in the top-left corner.

$\mathrm{Cofactors}(A)=\left(\begin{array}{lll}+& -& +\\ -& +& -\\ +& -& +\end{array}\right)$

So the matrix of cofactors is:

$\mathrm{Cofactors}(A)=\left(\begin{array}{ccc}ei-fh& -(di-fg)& dh-eg\\ -(bi-ch)& ai-cg& -(ah-bg)\\ bf-ce& -(af-cd)& ae-bd\end{array}\right)$

Step 3: Find the Adjoint (Adjugate) of the Matrix

The adjoint of A is the transpose of the matrix of cofactors:

$\mathrm{Adj}(A)={\left(\begin{array}{ccc}ei-fh& -(bi-ch)& bf-ce\\ -(di-fg)& ai-cg& -(af-cd)\\ dh-eg& -(ah-bg)& ae-bd\end{array}\right)}^T$

$\mathrm{Adj}(A)=\left(\begin{array}{ccc}ei-fh& -(di-fg)& dh-eg\\ -(bi-ch)& ai-cg& -(ah-bg)\\ bf-ce& -(af-cd)& ae-bd\end{array}\right)$

Step 4: Multiply by $\frac{1}{\det (A)}$

Finally, we need to calculate the determinant of \(A\):

$\det (A)=a(ei-fh)-b(di-fg)+c(dh-eg)$

Then, the inverse of A is:

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

By following these steps, we can find the inverse of any 3 × 3 matrix, provided that the determinant is non-zero.

3.0Example on How to Find Inverse of Matrix

Example: Find the Inverse of a 3 × 3 Matrix

Let us calculate the inverse of the following 3 × 3 matrix A:

$A=\left(\begin{array}{lll}1& 2& 3\\ 0& 1& 4\\ 5& 6& 0\end{array}\right)$

Step 1: Calculate the Matrix of Minors

1. Minor of a₁₁ = 1:

$M_{11}=\left|\begin{array}{ll}1& 4\\ 6& 0\end{array}\right|=(1\cdot 0-4\cdot 6)=-24$

2. Minor of a₁₂ = 2:

$M_{12}=\left|\begin{array}{ll}0& 4\\ 5& 0\end{array}\right|=(0\cdot 0-4\cdot 5)=-20$

3. Minor of a₁₃ = 3:

$M_{13}=\left|\begin{array}{ll}0& 1\\ 5& 6\end{array}\right|=(0\cdot 6-1\cdot 5)=-5$

4. Minor of a₂₁ = 0:

$M_{21}=\left|\begin{array}{ll}2& 3\\ 6& 0\end{array}\right|=(2\cdot 0-3\cdot 6)=-18$

5. Minor of a₂₂ = 1:

$M_{22}=\left|\begin{array}{ll}1& 3\\ 5& 0\end{array}\right|=(1\cdot 0-3\cdot 5)=-15$

6. Minor of a₂₃ = 4:

$M_{23}=\left|\begin{array}{ll}1& 2\\ 5& 6\end{array}\right|=(1\cdot 6-2\cdot 5)=-4$

7. Minor of a₃₁ = 5:

$M_{31}=\left|\begin{array}{ll}2& 3\\ 1& 4\end{array}\right|=(2\cdot 4-3\cdot 1)=5$

8. Minor of a₃₂ = 6:

$M_{32}=\left|\begin{array}{ll}1& 3\\ 0& 4\end{array}\right|=(1\cdot 4-3\cdot 0)=4$

9. Minor of a₃₃ = 0:

$M_{33}=\left|\begin{array}{ll}1& 2\\ 0& 1\end{array}\right|=(1\cdot 1-2\cdot 0)=1$

The matrix of minors is:

$\mathrm{Minors}(A)=\left(\begin{array}{ccc}-24& -20& -5\\ -18& -15& -4\\ 5& 4& 1\end{array}\right)$

Step 2: Convert to the Matrix of Cofactors

Apply the checkerboard pattern of signs:

$\mathrm{Cofactors}(A)=\left(\begin{array}{ccc}-24& 20& -5\\ 18& -15& 4\\ 5& -4& 1\end{array}\right)$

Step 3: Find the Adjoint (Adjugate) of the Matrix

The adjoint is the transpose of the matrix of cofactors:

$\mathrm{Adj}(A)=\left(\begin{array}{ccc}-24& 18& 5\\ 20& -15& -4\\ -5& 4& 1\end{array}\right)$

Step 4: Calculate the Determinant

The determinant of A is:

$\det (A)=1(1\cdot 0-4\cdot 6)-2(0\cdot 0-4\cdot 5)+3(0\cdot 6-1\cdot 5)$

$\det (A)=1(0-24)-2(0-20)+3(0-5)$

$\det (A)=-24+40-15$

$\det (A)=1$

Step 5: Multiply by $\frac{1}{\det (A)}$

Since $\det (A)=1$, the inverse is simply the adjoint matrix:

${\mathrm{A}}^{-1}=\frac{1}{1}\cdot \mathrm{Adj}(A)$

$A^{-1}=\left(\begin{array}{ccc}-24& 18& 5\\ 20& -15& -4\\ -5& 4& 1\end{array}\right)$

Thus, the inverse of matrix A is:

$A^{-1}=\left(\begin{array}{ccc}-24& 18& 5\\ 20& -15& -4\\ -5& 4& 1\end{array}\right)$

Here are some practice questions to help you master finding the inverse of a 3 × 3 matrix:

4.0Practice Questions on How to Find Inverse of Matrix

1. Matrix A: $A=\left(\begin{array}{ccc}2& 3& 1\\ 1& 0& -1\\ 3& 4& 2\end{array}\right)$
2. Matrix B: $B=\left(\begin{array}{lll}4& 7& 2\\ 3& 6& 1\\ 2& 5& 3\end{array}\right)$
3. Matrix C: $C=\left(\begin{array}{lll}1& 2& 3\\ 0& 1& 4\\ 5& 6& 0\end{array}\right)$
4. Matrix D: $D=\left(\begin{array}{ccc}1& 0& 2\\ -1& 3& 1\\ 3& 4& 0\end{array}\right)$
5. Matrix E: $\mathrm{E}=\left(\begin{array}{lll}2& 1& 1\\ 3& 2& 1\\ 2& 1& 2\end{array}\right)$

Answers:

1.  $A^{-1}=\left(\begin{array}{ccc}-\frac{4}{3}& \frac{2}{3}& -1\\ \frac{5}{3}& -\frac{1}{3}& -1\\ -\frac{4}{3}& -\frac{1}{3}& 1\end{array}\right)$
2. $B^{-1}=\left(\begin{array}{ccc}3& -17& 11\\ -3& 10& -6\\ 1& -2& 1\end{array}\right)$
3. $C^{-1}=\left(\begin{array}{ccc}-24& 18& 5\\ 20& -15& -4\\ -5& 4& 1\end{array}\right)$
4. ${\mathrm{D}}^{-1}=\left(\begin{array}{ccc}-4& 2& 1\\ 1& 0& 0\\ 5& -3& -1\end{array}\right)$
5. ${\mathrm{E}}^{-1}=\left(\begin{array}{ccc}3& -1& -1\\ -2& 2& 0\\ -1& 0& 1\end{array}\right)$

5.0Sample Questions on Find Inverse of Matrix

1. What is the inverse of a matrix?

Ans: The inverse of a matrix A is another matrix, denoted as ${\mathrm{A}}^{-1}$ , such that when A is multiplied by A^–1, the result is the identity matrix I. Mathematically, AA^–1 = A^–1A = I, where I is called the identity matrix.

2. When does a matrix have an inverse?

Ans: A matrix has an inverse if and only if it is a square matrix with the same number of rows and columns and its determinant is non-zero. Such a matrix is called non-singular or invertible.

3. How do you find the inverse of a 2 × 2 matrix?

Ans:  For a 2 × 2 matrix $A=\left(\begin{array}{ll}a& b\\ c& d\end{array}\right)$ the inverse is given by $A^{-1}=\frac{1}{ad-bc}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$, provided .

4. What are the steps to find the inverse of a 3 × 3 matrix?

Ans: To find the inverse of a 3 × 3 matrix, follow these steps:

1. Calculate the matrix of minors.
2. Convert the matrix of minors to the matrix of cofactors.
3. Find the adjoint (transpose of the matrix of cofactors).
4. Calculate the determinant of the original matrix.
5. Multiply the adjoint by $\frac{1}{\det (\mathrm{A})}.$

5. Can every square matrix be inverted?

Ans: No, only non-singular (invertible) square matrices can be inverted. A matrix is considered non-singular if its determinant is non-zero. Conversely, if the determinant is zero, the matrix is singular and does not have an inverse.

6. What is the identity matrix?

Ans: The identity matrix, denoted as I, is a square matrix with ones on the main diagonal and zeros elsewhere. For a 3 × 3 matrix, it looks like: $\mathrm{I}=\left(\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

Multiplying any matrix by the identity matrix leaves the original matrix unchanged.