NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.5

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NCERT Solutions for Class 12 Maths Chapter 4 Determinants Exercise 4.5

Ex 4.5 Class 12 NCERT Solutions Question 1.
\(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\)
Adj A = \(\begin{bmatrix} 4 & -3 \\ -2 & 1 \end{bmatrix}\)

Exercise 4.5 Class 12 NCERT Solutions Question 2.
\(\left[ \begin{matrix} 1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1 \end{matrix} \right]\)
Solution:

Exercise 4.5 Class 12 Maths Question 3.
\(\begin{bmatrix} 2 & 3 \\ -4 & 6 \end{bmatrix}\)
Solution:

4.5 Class 12 NCERT Solutions Question 4.
\(\left[ \begin{matrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{matrix} \right]\)
Solution:
|A| = \(\left|\begin{array}{ccc}
1 & -1 & 2 \\
3 & 0 & -2 \\
1 & 0 & 3
\end{array}\right|\)
A₁₁ = 0, A₁₂ = – 11, A₁₃ = 0,
A₂₁ = – 3, A₂₂ = 1, A₂₃ = 1,
A₃₁ = – 2, A₃₂ = 8, A₃₃ = 3

Ex 4.5 Class 12 Maths Ncert Solutions Question 5.
\(\begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}\)
Solution:

Exercise 4.5 Maths Class 12 Question 6.
\(\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}\)
Solution:

Exercise 4.5 Class 12 Maths Solutions Question 7.
\(\left[ \begin{matrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{matrix} \right]\)
Solution:

Class 12 Maths Ch 4 Ex 4.5 Question 8.
\(\left[ \begin{matrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{matrix} \right]\)
Solution:

Ex4.5 Class 12 NCERT Solutions Question 9.
\(\left[ \begin{matrix} 2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1 \end{matrix} \right]\)
Solution:

Ex 4.5 Class 12 Maths NCERT Solutions Question 10.
\(\left[ \begin{matrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{matrix} \right]\)
Solution:

Class 12 Maths Chapter 4 Exercise 4.5 Solutions Question 11.
\(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & cos\alpha & sin\alpha \\ 0 & sin\alpha & -cos\alpha \end{matrix} \right] \)
Solution:

Ncert Solutions For Class 12 Maths Chapter 4 Exercise 4.5 Question 12.
Let \(A=\begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix},B=\begin{bmatrix} 6 & 8 \\ 7 & 9 \end{bmatrix}\), verify that (AB)^-1 = B^-1A^-1
Solution:

Class 12 Maths Chapter 4 Exercise 4.5 Question 13.
If \(A=\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \) show that A² – 5A + 7I = 0, hence find A^-1.
Solution:

Class 12 Maths Chapter 4 Exercise 4.5 Solution Question 14.
For the matrix A = \(\begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix} \) find file numbers a and b such that A² + aA + bI² = 0. Hence, find A^-1.
Solution:

Class 12 Maths Ex 4.5 Solutions Question 15.
For the matrix \(A=\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{matrix} \right] \) Show that A³ – 6A² + 5A + 11I₃=0. Hence find A^-1
Solution:

i.e., A³ – 6A² + 5A + 11I₃ = 0 … (1)
|A| = \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & 2 & -3 \\
2 & -1 & 3
\end{array}\right|\)
= 1(3) – 1(9) + 1(- 5) = – 11 ≠ 0
∴ A^-1 exists
⇒ (1) = A.A.A – 6A.A. + 5A = – 11 I
⇒ AA(AA^-1 – 6A(AA^-1) + 5AA^-1 = – 11 IA^-1 (Multiplying by A^-1)
⇒ A²I – 6AI + 5I = – 11A^-1
⇒ A² – 6A + 5I = – 11A^-1
⇒ A^-1 = \(\frac { -1 }{ 11 }\)[A² – 6A + 5I]

Class 12 Ex 4.5 NCERT Solutions Question 16.
If \(A=\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right] \) show that A³ – 6A² + 9A – 4I = 0 and hence, find A^-1
Solution:

Exercise 4.5 Class 12 Maths Ncert Solutions Question 17.
Let A be a non-singular square matrix of order 3 x 3. Then | Adj A | is equal to:
(a) | A |
(b) | A |²
(c) | A |³
(d) 3 | A |
Solution:

Hence option (b) is correct.

Class 12 Maths Ex 4.5 NCERT Solutions Question 18.
If A is an invertible matrix of order 2, then det. (A^-1) is equal to:
(a) det. (A)
(b) \(\\ \frac { 1 }{ det.(A) } \)
(c) 1
(d) 0
Solution:
|A|≠0
⇒ A^-1 exists ⇒ A^-1 = I
|AA^-1| = |I| = I
⇒ |A||A^-1| = I
\(|{ A }^{ -1 }|=\frac { 1 }{ |A| } \)
Hence option (b) is correct.