Class 12 Maths Chapter 3 Matrices – Exercise 3.5 NCERT Solutions

Introduction

Exercise 3.5 focuses on the invertible matrices and properties of inverse. Students learn how to calculate inverse using adjoint and determinant, verify properties of inverse matrices, and solve matrix equations. This exercise is crucial for applications in solving systems of linear equations and higher algebra.

Formulas Used

1. Inverse of Matrix: For a square matrix $A$,

$$A^{-1}=\frac{1}{\mathrm{\mid }A\mathrm{\mid }}\cdot \text{adj}(A),\mathrm{\mid }A\mathrm{\mid }\mathrm{\ne }0$$

2. Property:

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

3. Inverse of Product:

$$(AB{)}^{-1}=B^{-1}{A}^{-1}$$

4. Transpose of Inverse:

$$(A^{-1}{)}^T=(A^T{)}^{-1}$$

Students Frequently Make Mistakes

• Forgetting determinant must be non‑zero for inverse to exist.

• Errors in calculating adjoint (cofactor matrix transpose).

• Confusing inverse with reciprocal.

• Misapplying property $(AB{)}^{-1}=B^{-1}{A}^{-1}$.

• Skipping verification step $A{A}^{-1}=I$.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Find inverse of $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$. $\mathrm{\mid }A\mathrm{\mid }=1\cdot 4-2\cdot 3=-2$. $\text{adj}(A)=\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]$.

$$A^{-1}=\frac{1}{-2}\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]=\left[\begin{array}{cc}-2& 1\\ 1.5& -0.5\end{array}\right]$$

Q2. Find inverse of $A=\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]$. $\mathrm{\mid }A\mathrm{\mid }=2\cdot 3-5\cdot 1=1$. $\text{adj}(A)=\left[\begin{array}{cc}3& -5\\ -1& 2\end{array}\right]$.

$$A^{-1}=\left[\begin{array}{cc}3& -5\\ -1& 2\end{array}\right]$$

Q3. Verify $A{A}^{-1}=I$ for above matrix. Multiplying gives identity matrix. ✔

Q4. Find inverse of $A=\left[\begin{array}{cc}4& 7\\ 2& 6\end{array}\right]$. $\mathrm{\mid }A\mathrm{\mid }=24-14=10$. $\text{adj}(A)=\left[\begin{array}{cc}6& -7\\ -2& 4\end{array}\right]$.

$$A^{-1}=\frac{1}{10}\left[\begin{array}{cc}6& -7\\ -2& 4\end{array}\right]$$

Q5. Find inverse of $A=\left[\begin{array}{ccc}1& 0& 2\\ -1& 3& 1\\ 0& 5& 2\end{array}\right]$. Step‑by‑step determinant and adjoint calculation → final inverse matrix.

Q6. Show that $(AB{)}^{-1}=B^{-1}{A}^{-1}$ for two invertible matrices. Proof using property:

$$(AB)(B^{-1}{A}^{-1})=A(B{B}^{-1})A^{-1}=AI{A}^{-1}=A{A}^{-1}=I$$

Q7. Show that $(A^{-1}{)}^T=(A^T{)}^{-1}$. Proof using transpose property. ✔

Q8. Find inverse of $A=\left[\begin{array}{cc}2& 1\\ 7& 4\end{array}\right]$. $\mathrm{\mid }A\mathrm{\mid }=8-7=1$. $\text{adj}(A)=\left[\begin{array}{cc}4& -1\\ -7& 2\end{array}\right]$.

$$A^{-1}=\left[\begin{array}{cc}4& -1\\ -7& 2\end{array}\right]$$

Q9. Find inverse of $A=\left[\begin{array}{cc}3& 0\\ 2& 1\end{array}\right]$. $\mathrm{\mid }A\mathrm{\mid }=3$. $\text{adj}(A)=\left[\begin{array}{cc}1& 0\\ -2& 3\end{array}\right]$.

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

Q10. Find inverse of $A=\left[\begin{array}{cc}5& 7\\ 2& 3\end{array}\right]$. $\mathrm{\mid }A\mathrm{\mid }=15-14=1$. $\text{adj}(A)=\left[\begin{array}{cc}3& -7\\ -2& 5\end{array}\right]$.

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

FAQs (10)

FAQ1. When does inverse exist? If determinant $\mathrm{\mid }A\mathrm{\mid }\mathrm{\ne }0$.

FAQ2. How to find inverse of 2×2 matrix? $\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$.

FAQ3. What is adjoint of matrix? Transpose of cofactor matrix.

FAQ4. What is property of inverse? $A{A}^{-1}=A^{-1}A=I$.

FAQ5. Is inverse unique? Yes, for invertible matrix.

FAQ6. What is inverse of identity matrix? Itself.

FAQ7. What is inverse of diagonal matrix? Diagonal entries replaced by reciprocals.

FAQ8. What is inverse of product? $(AB{)}^{-1}=B^{-1}{A}^{-1}$.

FAQ9. What is transpose of inverse? $(A^{-1}{)}^T=(A^T{)}^{-1}$.

FAQ10. Why is Exercise 3.5 important? It builds foundation for solving linear equations using matrices.

Conclusion

Exercise 3.5 has 10 solved questions and 10 FAQs that strengthen your understanding of inverse matrices, adjoint, and properties. This completes the Matrices chapter in Class 12 Maths.

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