Solved Examples on Inverse of 2×2 Matrix

Example 1: Find the inverse of matrix B = [Tex]\begin{bmatrix} 10 & 5\\ 7 & 3 \end{bmatrix}[/Tex] using inverse formula

Solution:

B = [Tex]\begin{bmatrix} 10 & 5\\ 7 & 3 \end{bmatrix} [/Tex]

The inverse of 2×2 matrix formula is given by:

B^-1 = [Tex]\bold{\frac{1}{ad – bc} \begin{bmatrix} d &-b\\ -c & a \end{bmatrix}}[/Tex]

⇒ B^-1 = [Tex]\frac{1}{(10\times 3) – (7\times 5)} \begin{bmatrix} 3 &-5\\ -7 & 10 \end{bmatrix}[/Tex]

⇒ B^-1 = [Tex]\frac{1}{ 30 – 35} \begin{bmatrix} 3 &-5\\ -7 & 10 \end{bmatrix} [/Tex]

⇒ B^-1 = [Tex]\frac{1}{-5} \begin{bmatrix} 3 &-5\\ -7 & 10 \end{bmatrix} [/Tex]

Example 2: Find the inverse of matrix C = [Tex]\begin{bmatrix} 30 & 8\\ 7 & 1 \end{bmatrix}[/Tex] using inverse formula.

Solution:

C = [Tex]\begin{bmatrix} 30 & 8\\ 7 & 1 \end{bmatrix} [/Tex]

The inverse of 2×2 matrix formula is given by:

C^-1 = [Tex] \bold{\frac{1}{ad – bc} \begin{bmatrix} d &-b\\ -c & a \end{bmatrix}}[/Tex]

⇒ C^-1 = [Tex]\frac{1}{(30\times 1) – (7\times 8)} \begin{bmatrix} 1 &-7\\ -8 & 30 \end{bmatrix}[/Tex]

⇒ C^-1 = [Tex]\frac{1}{30 – 56} \begin{bmatrix} 1 &-7\\ -8 & 30 \end{bmatrix}[/Tex]

⇒ C^-1 = [Tex]\frac{1}{ – 26} \begin{bmatrix} 1 &-7\\ -8 & 30 \end{bmatrix}[/Tex]

Example 3: Find the inverse of 2×2 matrix D = [Tex]\begin{bmatrix} 1& 4\\ 6 & 12 \end{bmatrix}[/Tex] using elementary operations method.

Solution:

To find the inverse of D = [Tex]\begin{bmatrix} 1& 4\\ 6 & 12 \end{bmatrix}[/Tex] we will use elementary row operations.

D = ID

⇒ [Tex]\begin{bmatrix} 1& 4\\ 6 & 12 \end{bmatrix} [/Tex]= [Tex]\begin{bmatrix} 1& 0\\ 0 & 1 \end{bmatrix} [/Tex]D

R₂ → R₂ – 6R₁

⇒ [Tex]\begin{bmatrix} 1& 4\\ 0 & -12 \end{bmatrix}[/Tex] = [Tex]\begin{bmatrix} 1& 0\\ – 6& 1 \end{bmatrix}[/Tex] D

R₂ → R₂ / (-12)

⇒ [Tex]\begin{bmatrix} 1& 4\\ 0 & 1 \end{bmatrix}[/Tex] = [Tex]\begin{bmatrix} 1& 0\\ 1/2 & -1/12 \end{bmatrix}[/Tex] D

R₁ → R₁ – 4R₂

⇒ [Tex]\begin{bmatrix} 1& 0\\ 0 & 1 \end{bmatrix} [/Tex] = [Tex]\begin{bmatrix} -1& 1/3\\ 1/2 & -1/12 \end{bmatrix} [/Tex] D

D^-1 = [Tex] \begin{bmatrix} -1& 1/3\\ 1/2 & -1/12 \end{bmatrix} [/Tex]

Example 4: Find the inverse of 2×2 matrix E = [Tex]\begin{bmatrix} 1& 5\\ 8 & 2 \end{bmatrix}[/Tex] using elementary operations method.

Solution:

To find the inverse of E = [Tex] \begin{bmatrix} 1& 5\\ 8 & 2 \end{bmatrix} [/Tex]we will use elementary row operation.

E = IE

⇒ [Tex] \begin{bmatrix} 1& 5\\ 8 & 2 \end{bmatrix} [/Tex] = [Tex] \begin{bmatrix} 1& 0\\ 0 & 1 \end{bmatrix} [/Tex] E

R₂ → R₂ – 8R₁

⇒ [Tex] \begin{bmatrix} 1& 5\\ 0 & -38 \end{bmatrix} [/Tex]= [Tex] \begin{bmatrix} 1& 0\\ -8 & 1 \end{bmatrix} [/Tex] E

R₂ → R₂ / (-38)

⇒ [Tex] \begin{bmatrix} 1& 5\\ 0 & 1 \end{bmatrix} [/Tex] = [Tex]\begin{bmatrix} 1& 0\\ 8/38 & -1/38 \end{bmatrix}[/Tex] E

R₁ → R₁ – 5R₂

⇒ [Tex] \begin{bmatrix} 1& 0\\ 0 & 1 \end{bmatrix}[/Tex] = [Tex]\begin{bmatrix} -2/38& 5/38\\ 8/38 & -1/38 \end{bmatrix}[/Tex] E

So, the inverse of matrix E i.e., E^-1 = [Tex] \begin{bmatrix} -2/38& 5/38\\ 8/38 & -1/38 \end{bmatrix} [/Tex]

Inverse of 2×2 matrix is the matrix obtained by dividing the adjoint of the matrix by the determinant of the matrix. The two methods to find the inverse of 2×2 matrix is by using inverse formula and by using elementary operations.

In this article, we will explore how to find the inverse of 2×2 matrix along with both the methods and basics of the inverse of matrix. We will also solve some examples of the inverse of 2×2 matrix. Let’s start our learning on the topic “How to Find the Inverse of 2×2 Matrix?”.