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QUESTION: 1

Find the value of x for the singular matrix?

Solution:

Determinant of a singular matrix is 0

∴ 3×15×(24−3x)=0

∴ 3x=24

∴ x=8

*Answer can only contain numeric values

QUESTION: 2

If the determinant of the matrix is 26, then the determinant of the matrix is

Solution:

If two rows are interchanged in a determinant then the value of the determinant does not change but sign will change. In the given question, first and third rows are interchanged.

Determinant wil be -26

QUESTION: 3

The value of the determinant

Solution:

QUESTION: 4

If a matrix A is given by f(x) = a_{0} + a_{1}x + a_{2}x^{2}+…+ a_{n−1}x^{n−1} + a_{n}x^{n}, then the determinant of A is

Solution:

matrix ‘A’ ⇒ f(x) = a_{0 }+ a_{1}x + a_{2}x^{2 }+……….+a_{n−1}x^{n−1 }+ a_{n}x^{n}f(x)

if 2 × 2 matrix then characteristic equation ⇒ aλ^{2 }+ bλ + c = 0

⇒ |A| = λ_{1}.λ_{2}. = c/a

where a_{o} = c;a_{n} = a_{2} = a

if 3 × 3 matrix → then characteristic equation ⇒ aλ^{3 }+ bλ^{2 }+ cλ + d = 0

∴ For n × n matrix, |A| = λ_{1}.λ_{2}.λ_{3}….λ_{n}

QUESTION: 5

What is the determinant of the below-given matrix?

Solution:

Determinant of the given matrix is

Δ = (a − b) (a − c) (a − d) (b − c) (b − d)(c − d)

a = 3, b = 5, c = 7, d = 9

∴ Δ = 768

QUESTION: 6

If A = then the value of adj (adj (A))?

Solution:

A is upper triangular matrix

∴ product of diagonal elements = |A|

= 1 × 5 × 8 × 10 = 400

Note:

|A||A| is determinant of matrix A

*Answer can only contain numeric values

QUESTION: 7

What is the rank of A−B where

Solution:

∴ rank is < 3

Also

**Hence rank of is A−B is 2**

QUESTION: 8

What is the value of A^{3 }− 9A^{2 }− 47A?

Solution:

Characteristic equation of the given matrix is

Every matrix satisfies its own characteristic equation

∴A^{3} − 9A^{2} − 47A = 20I

QUESTION: 9

The factorized form of the following determinant is:

Solution:

Applying R2 → R2 – R1

Now, expanding from a11.

= (m - l)(n - l)(n + l - m - l)

= (m - l)(n - l)(n - m)

QUESTION: 10

A 3 × 5 matrix has all its entries equal to 1. The rank of the matrix is

Solution:

Rank ≤ 3 since all entries are 1,

∴| 3 × 3 submatrices | = | 2 × 2 submatrices | = 0

only 1 × 1 or single entries are ≠ 0

∴ rank = 1

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