# Mathematics Questions and Answers – Determinants – Adjoint and Inverse of a Matrix

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This set of Mathematics Problems focuses on “Determinants – Adjoint and Inverse of a Matrix”.

1. Which of the following is the adjoint of the matrix A=$$\begin{bmatrix}1&5\\3&4\end{bmatrix}$$?
a) $$\begin{bmatrix}4&-5\\-3&-1\end{bmatrix}$$
b) $$\begin{bmatrix}-4&5\\-3&1\end{bmatrix}$$
c) $$\begin{bmatrix}4&-5\\-3&1\end{bmatrix}$$
d) $$\begin{bmatrix}4&5\\-3&1\end{bmatrix}$$

Explanation: We have A11=(-1)1+1 4=4
A12=(-1)1+2 3=-3
A21=(1)2+1 5=-5
A22=(-1)2+2 1=1
∴adj A=$$\begin{bmatrix}A_{11}&A_{21}\\A_{12}&A_{22}\end{bmatrix}$$=$$\begin{bmatrix}4&-5\\-3&1\end{bmatrix}$$.

2. If A=$$\begin{bmatrix}5&-8\\2&6\end{bmatrix}$$, find A(adj A).
a) $$\begin{bmatrix}41&0\\0&46\end{bmatrix}$$
b) $$\begin{bmatrix}46&0\\1&46\end{bmatrix}$$
c) $$\begin{bmatrix}46&1\\0&46\end{bmatrix}$$
d) $$\begin{bmatrix}46&0\\0&46\end{bmatrix}$$

Explanation: Given that, A=$$\begin{bmatrix}5&-8\\2&6\end{bmatrix}$$
∴adj A=$$\begin{bmatrix}6&8\\-2&5\end{bmatrix}$$
A(adj A)=$$\begin{bmatrix}5&-8\\2&6\end{bmatrix}\begin{bmatrix}6&8\\-2&5\end{bmatrix}$$
=$$\begin{bmatrix}5×6+(-8)×(-2)&5×8+5×(-8)\\2×6+6×(-2)&2×8+6×5\end{bmatrix}$$=$$\begin{bmatrix}46&0\\0&46\end{bmatrix}$$.

3. If A=$$\begin{bmatrix}1&0\\9&4\end{bmatrix}$$, then (adj A)A is ______________
a) $$\begin{bmatrix}-4&0\\0&-4\end{bmatrix}$$
b) $$\begin{bmatrix}4&0\\1&4\end{bmatrix}$$
c) $$\begin{bmatrix}4&0\\0&4\end{bmatrix}$$
d) $$\begin{bmatrix}4&0\\0&-4\end{bmatrix}$$

Explanation: Given that, A=$$\begin{bmatrix}1&0\\9&4\end{bmatrix}$$
∴|A|=4-0=4
⇒A(adj A)=|A|I=$$\begin{bmatrix}4&0\\0&4\end{bmatrix}$$.

4. Which of the following is the formula for calculating the inverse of the matrix?
a) $$\frac{2}{|A|}$$ adj A
b) $$\frac{1}{|A|}$$ adj A
c) $$\frac{-1}{|A|}$$ adj A
d) $$\frac{1}{|2A|}$$ adj A

Explanation: The formula for calculating the inverse of the matrix is given by
A-1=$$\frac{1}{|A|}$$ adj A, where |A| is the determinant of the matrix and adj A is the adjoint of the matrix.

5. Find the inverse of the matrix A=$$\begin{bmatrix}8&5\\4&1\end{bmatrix}$$.
a) $$\begin{bmatrix}-\frac{1}{12}&\frac{5}{12}\\\frac{1}{3}&-\frac{2}{3}\end{bmatrix}$$
b) $$\begin{bmatrix}\frac{1}{12}&\frac{5}{12}\\\frac{1}{3}&-\frac{2}{3}\end{bmatrix}$$
c) $$\begin{bmatrix}-\frac{1}{12}&\frac{5}{12}\\\frac{1}{3}&\frac{2}{3}\end{bmatrix}$$
d) $$\begin{bmatrix}-\frac{1}{12}&\frac{5}{12}\\-\frac{1}{3}&-\frac{2}{3}\end{bmatrix}$$

Explanation: Give that, A=$$\begin{bmatrix}8&5\\4&1\end{bmatrix}$$
adj A=$$\begin{bmatrix}1&-5\\-4&8\end{bmatrix}$$
|A|=8×1-(-5)×(-4)=8-20=-12
A-1=$$\frac{1}{|A|}$$ adj A=$$\frac{1}{-12} \begin{bmatrix}1&-5\\-4&8\end{bmatrix}$$=$$\begin{bmatrix}-\frac{1}{12}&\frac{5}{12}\\\frac{1}{3}&-\frac{2}{3}\end{bmatrix}$$.

6. Which of the below condition is incorrect for the inverse of a matrix A?
a) The matrix A must be a square matrix
b) A must be singular matrix
c) A must be a non-singular matrix

Explanation: The matrix should not be a singular matrix. A square matrix is said to be singular |A|=0.
We know that, A-1=$$\frac{1}{|A|}$$ adj A,
Hence, if |A|=0 the inverse of the matrix does not exist.

7. Which of the below given matrices has the inverse $$\frac{1}{-6}\begin{bmatrix}2&1\\0&-3\end{bmatrix}$$?
a) $$\begin{bmatrix}3&-1\\0&2\end{bmatrix}$$
b) $$\begin{bmatrix}-3&-1\\0&2\end{bmatrix}$$
c) $$\begin{bmatrix}-2&0\\1&3\end{bmatrix}$$
d) $$\begin{bmatrix}-3&-1\\0&-2\end{bmatrix}$$

Explanation: Consider the matrix $$\begin{bmatrix}-3&-1\\0&2\end{bmatrix}$$
adj A=$$\begin{bmatrix}2&1\\0&-3\end{bmatrix}$$
|A|=-6
∴A-1=$$\frac{1}{|A|}$$ adj A=$$\frac{1}{-6}\begin{bmatrix}2&1\\0&-3\end{bmatrix}$$.

8. If A=$$\begin{bmatrix}-8&2\\6&-3\end{bmatrix}$$ and B=$$\begin{bmatrix}2&1\\1&7\end{bmatrix}$$. Find (AB)-1.
a) –$$\frac{1}{432}$$ $$\begin{bmatrix}-27&6\\9&14\end{bmatrix}$$
b) $$\frac{1}{432}$$ $$\begin{bmatrix}27&6\\9&14\end{bmatrix}$$
c) $$\frac{1}{432}$$ $$\begin{bmatrix}-27&6\\9&14\end{bmatrix}$$
d) $$\frac{-1}{432}$$ $$\begin{bmatrix}27&6\\9&14\end{bmatrix}$$

Explanation: Given that, A=$$\begin{bmatrix}-8&2\\6&-3\end{bmatrix}$$ and B=$$\begin{bmatrix}2&1\\1&7\end{bmatrix}$$
∴AB=$$\begin{bmatrix}-8×2+2×1&-8×1+2×7\\6×2+(-3)×1&6×1+(-3)×7\end{bmatrix}$$=$$\begin{bmatrix}-14&6\\9&27\end{bmatrix}$$
adj(AB)=$$\begin{bmatrix}27&-6\\-9&-14\end{bmatrix}$$
|AB|=27×(-14)-(-9)×(-6)=-378-54=-432
(AB)-1=$$\frac{1}{|AB|}$$ adj AB=$$\frac{1}{-432} \begin{bmatrix}27&-6\\-9&-14\end{bmatrix}$$=$$\frac{1}{432} \begin{bmatrix}-27&6\\9&14\end{bmatrix}$$.

9. Which of the following formula is incorrect?
b) |adj (A)|=|A|n-1, for an nth order matrix
c) A-1=$$\frac{1}{|A|}$$ adj A

10. A square matrix A is said to be non-singular if |A|≠0.
a) True
b) False

Explanation: The given statement is true. A square matrix A is said to be singular if |A|=0 and non-singular if A≠0.

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

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