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) \(\begin{bmatrix}4&5\\-3&1\end{bmatrix}\)
View Answer

Answer: c
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=\(\begin{bmatrix}A_{11}&A_{21}\\A_{12}&A_{22}\end{bmatrix}\)=\(\begin{bmatrix}4&-5\\-3&1\end{bmatrix}\).
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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) \(\begin{bmatrix}46&0\\0&46\end{bmatrix}\)
View Answer

Answer: d
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) \(\begin{bmatrix}4&0\\0&-4\end{bmatrix}\)
View Answer

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

Answer: b
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) \(\begin{bmatrix}-\frac{1}{12}&\frac{5}{12}\\-\frac{1}{3}&-\frac{2}{3}\end{bmatrix}\)
View Answer

Answer: a
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=\(\frac{1}{-12} \begin{bmatrix}1&-5\\-4&8\end{bmatrix}\)=\(\begin{bmatrix}-\frac{1}{12}&\frac{5}{12}\\\frac{1}{3}&-\frac{2}{3}\end{bmatrix}\).
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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
d) adj A≠0
View Answer

Answer: b
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) \(\begin{bmatrix}-3&-1\\0&-2\end{bmatrix}\)
View Answer

Answer: b
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=\(\frac{1}{-6}\begin{bmatrix}2&1\\0&-3\end{bmatrix}\).
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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) \(\frac{-1}{432}\) \(\begin{bmatrix}27&6\\9&14\end{bmatrix}\)
View Answer

Answer: c
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?
a) A(adj A)=|A|I
b) |adj (A)|=|A|n-1, for an nth order matrix
c) A-1=\(\frac{1}{|A|}\) adj A
d) A(adj A)=|A|n-1
View Answer

Answer: d
Explanation: The formula A(adj A)=|A|n-1 is incorrect. The correct formula is A(adj A)=(adjA)A=|A|I.
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10. A square matrix A is said to be non-singular if |A|≠0.
a) True
b) False
View Answer

Answer: a
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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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter