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# Class 12 RD Sharma Solutions – Chapter 5 Algebra of Matrices – Exercise 5.3 | Set 2

• Last Updated : 21 Jul, 2021

### Question 26. If= 0, find x.

Solution:

We have,

=>= 0

=>

=>

=>

=> 2x â€“ 4 = 0

=> 2x = 4

=> x = 2

Therefore, the value of x is 2.

### Question 27. If A =and I =, then prove that A2 â€“ A + 2I = 0.

Solution:

We have,

A =

A2 =

=

=

L.H.S. = A2 â€“ A + 2I

=

=

=

=

=

=

= 0

= R.H.S.

Hence proved.

### Question 28. If A =and I =, then find Î» so that A2 = 5A + Î»I.

Solution:

We have,

A =

A2 =

=

=

We are given,

=> A2 = 5A + Î»I

=>

=>

=>

=>

On comparing both sides, we get

=> 8 = 15 + Î»

=> Î» = â€“7

Therefore, the value of Î» is â€“7.

### Question 29. If A =, show that A2 â€“ 5A + 7I2 = 0.

Solution:

We have,

A =

A2 =

=

=

L.H.S. = A2 â€“ 5A + 7I2

=

=

=

=

= 0

= R.H.S.

Hence proved.

### Question 30. If A =, show that A2 â€“ 2A + 3I2 = 0.

Solution:

We have,

A =

A2 =

=

=

L.H.S. = A2 â€“ 2A + 3I2

=

=

=

=

= 0

= R.H.S.

Hence proved.

### Question 31. Show that the matrix A =satisfies the equation A3 â€“ 4A2 + A = 0.

Solution:

We have,

A =

A2 =

=

=

A3 = A2. A

=

=

=

L.H.S. = A3 â€“ 4A2 + A

=

=

=

=

= 0

= R.H.S.

Hence proved.

### Question 32. Show that the matrix A =is root of the equation A2 â€“ 12A â€“ I = 0

Solution:

We have,

A =

A2 =

=

=

L.H.S. = A2 â€“ 12A â€“ I

=

=

=

=

= 0

= R.H.S.

Hence proved.

### Question 33. If A =find A2 â€“ 5A â€“ 14I.

Solution:

We have,

A =

A2 =

=

=

A2 â€“ 5A â€“ 14I =

=

=

=

### Question 34. If A =, find A2 â€“ 5A + 7I = 0. Use this to find A4.

Solution:

We have,

A =

A2 =

=

=

L.H.S. = A2 â€“ 5A + 7I = 0

=

=

=

=

= 0

= R.H.S.

Hence proved.

Now we have A2 â€“ 5A + 7I = 0

=> A2 = 5A â€“ 7I

=> A4 = (5A â€“ 7I) (5A â€“ 7I)

=> A4 = 25A2 â€“ 35AI â€“ 35AI + 49I

=> A4 = 25A2 â€“ 70AI + 49I

=> A4 = 25 (5A â€“ 7I) â€“ 70AI + 49I

=> A4 = 125A â€“ 175I â€“ 70A + 49I

=> A4 = 55A â€“ 126I

=> A4 =

=> A4 =

=> A4 =

=> A4 =

### Question 35. If A =, find k such that A2 = kA â€“ 2I2.

Solution:

We have,

A =

A2 =

=

=

We are given,

=> A2 = kA â€“ 2I2

=>

=>

=>

On comparing both sides, we get

=> 3k â€“ 2 = 1

=> 3k = 3

=> k = 1

Therefore, the value of k is 1.

### Question 36. If A =, find k such that A2 â€“ 8A + kI = 0.

Solution:

We have,

A =

A2 =

=

=

We are given,

=> A2 â€“ 8A + kI = 0

=>

=>

=>

=>

On comparing both sides, we get

=> â€“k + 7 = 0

=> k = 7

Therefore, the value of k is 7.

### Question 37. If A =and f(x) = x2 â€“ 2x â€“ 3, show that f(A) = 0.

Solution:

We have,

A =and f(x) = x2 â€“ 2x â€“ 3

A2 =

=

=

L.H.S. = f(A) = A2 â€“ 2A â€“ 3I2

=

=

=

=

= 0

= R.H.S.

Hence proved.

### Question 38. If A =and I =, find Î», Î¼ so that A2 = Î»A + Î¼I.

Solution:

We have,

A =

A2 =

=

=

We are given,

=> A2 = Î»A + Î¼I

=>

=>

=>

=>

On comparing both sides, we get,

=> 2Î» + Î¼ = 7 and Î» = 4

=> 2(4) + Î¼ = 7

=> Î¼ = 7 â€“ 8

=> Î¼ = â€“1

Therefore, the value of Î» is 4 and Î¼ is â€“1.

### Question 39. Find the value of x for which the matrix productequals an identity matrix.

Solution:

We have,

=>

=>

=>

On comparing both sides, we get,

=> 5x = 1

=> x = 1/5

Therefore, the value of x is 1/5.

### Question 40. Solve the following matrix equations:

(i)

Solution:

We have,

=>

=>

=>

=>

=> x2 â€“ 2x â€“ 15 = 0

=> x2 â€“ 5x + 3x â€“ 15 = 0

=> x (x â€“ 5) + 3 (x â€“ 5) = 0

=> (x â€“ 5) (x + 3) = 0

=> x = 5 or â€“3

Therefore, the value of x is 5 or â€“3.

(ii)

Solution:

We have,

=>

=>

=>

=>

=> 4 + 4x = 0

=> 4x = â€“4

=> x = â€“1

Therefore, the value of x is â€“1.

(iii)

Solution:

We have,

=>

=>

=>

=>

=> x2 â€“ 48 = 0

=> x2 = 48

=> x = Â±4âˆš3

Therefore, the value of x is Â±4âˆš3.

(iv)

Solution:

We have,

=>

=>

=>

=>

=> 2x2 + 23x = 0

=> x (2x + 23) = 0

=> x = 0 or x = â€“23/2

Therefore, the value of x is 0 or â€“23/2.

### Question 41. If A =, compute A2 â€“ 4A + 3I3.

Solution:

We have,

A =

A2 =

=

=

So, A2 â€“ 4A + 3I3 =

=

=

=

### Question 42. If f(x) = x2 â€“ 2x, find f(A), where A =.

Solution:

We have,

A =and f(x) = x2 â€“ 2x

A2 =

=

=

So, f(A) = A2 â€“ 2A

=

=

=

=

### Question 43. If f(x) = x3 + 4x2 â€“ x, find f(A) where A =.

Solution:

We have,

A =and f(x) = x3 + 4x2 â€“ x

A2 =

=

=

A3 = A2. A

=

=

=

Now, f(A) = A3 + 4A2 â€“ A

=

=

=

=

### Question 44. If A =, then show that A is a root of the polynomial f(x) = x3 â€“ 6x2 + 7x +2.

Solution:

We have,

A =and f(x) = x3 â€“ 6x2 + 7x +2.

A2 =

=

=

A3 = A2. A

=

=

=

In order to show that A is a root of above polynomial, we need to prove that f(A) = 0.

Now, f(A) = A3 â€“ 6A2 + 7A + 2I

=

=

=

=

= 0

Hence proved.

### Question 45. If A =, prove that A2 â€“ 4A â€“ 5I = 0.

Solution:

We have,

A =

A2 =

=

=

Now, L.H.S. = A2 â€“ 4A â€“ 5I

=

=

=

=

= 0

= R.H.S.

Hence proved.

### Question 46. If A =, show that A2 â€“ 7A + 10I3 = 0.

Solution:

We have,

A =

A2 =

=

=

Now, L.H.S. = A2 â€“ 7A + 10I3

=

=

=

=

= 0

= R.H.S.

Hence proved.

### Question 47. Without using the concept of inverse of a matrix, find the matrixsuch that,

Solution:

We have,

=>

=>

On comparing both sides, we get,

5x â€“ 7z = â€“16

5y â€“ 7u = â€“6

â€“2x + 3z = 7

â€“2y + 3u = 2

On solving the above equations, we get

=> x = 1, y = â€“4, z = 3 and u = â€“2.

So, we get.

### Question 48. Find the matrix A such that

(i)

Solution:

Let A =

Given equation is,

=>

=>

=>

=>

On comparing both sides, we get, a = 1, b = 0 and c = 1.

And x + 1 = 3 => x = 2

Also, y = 3 and

z + 1 = 5 => z = 4

So, we have A =

(ii)

Solution:

Let A =

Given equation is,

=>

=>

=>

On comparing both sides, we get,

w + 4x = 7

2w + 5x = â€“6

y + 4z = 2

2y + 5z = 4

On solving the above equations, we get

=> x = â€“2, y = 2, w = 1 and z = 0.

So, we get A =

(iii)

Solution:

Let A =

Given equation is,

=>

=>

=>

On comparing both sides, we get,

=> 4x = â€“ 4, 4y = 8 and 4z = 4.

=> x = â€“1, y = 2 and z = 1.

So, we get A =

(iv)

Solution:

We have,

A =

A =

A =

A =

(v)

Solution:

Let A =

Given equation is,

=>

=>

=>

On comparing both sides, we get,

=> x = 1, y = â€“2 and z = â€“5

And also we have,

2x â€“ a = â€“1

2y â€“ b = â€“8

2z â€“ c = â€“10

On solving these, we get,

=> a = 3, b = 4 and c = 0.

So, we get A =

(vi)

Solution:

Let A =

Given equation is,

=>

=>

=>

On comparing both sides, we get

x + 4a = â€“7 and 2x + 5a = â€“8

=> x = 1 and a = â€“2

y + 4b = 2 and 2y + 5b = 4

=> b = 0 and y = 2

z + 4c = 11 and 2z + 5c = 10

=> c = 4 and z = â€“5

So, we get A =

### Question 49. Find a 2 Ã— 2 matrix A such that= 6I2.

Solution:

Let A =

Given equation is,

=>= 6I

=>

=>

=>

On comparing both sides, we get

w + x = 6 and â€“2w + 4x = 0

=> w = 4 and x = 2

y + z = 0 and â€“2y + 4z = 6

=> y = â€“1 and z = 1

So, we get A =

### Question 50. If A =, find A16.

Solution:

We have,

A =

A2 =

=

=

A16 = A2 A2 A2 A2

=

=

### Question 51. If A =, B =and x2 = â€“1, then show that (A + B)2 = A2 + B2.

Solution:

We have,

A =, B =and x2 = â€“1

L.H.S. = (A + B)2

=

=

=

=

=

=

=

=

R.H.S. = A2 + B2

=

=

=

=

=

=

= L.H.S.

Hence proved.

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