GATE Computer Science Engineering(CSE) 2027 Test: Engineering Mathematics-

Determinant of a singular matrix is 0

∴ 3×15×(24−3x)=0
∴ 3x=24
∴ x=8

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

matrix ‘A’ ⇒ f(x) = a₀ + a₁x + a₂x² +……….+a_n−1xⁿ⁻¹ + a_nxⁿf(x)

if 2 × 2 matrix then characteristic equation  ⇒ aλ² + bλ + c = 0
⇒ |A| = λ₁.λ₂. = c/a
where a_o = c;a_n = a₂ = a 
if 3 × 3 matrix → then characteristic equation ⇒ aλ³ + bλ² + cλ + d = 0

∴ For n × n matrix, |A| = λ₁.λ₂.λ₃….λ_n

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

A is upper triangular matrix

∴ product of diagonal elements = |A|
 = 1 × 5 × 8 × 10 = 400

 

Note:

|A||A| is determinant of matrix A

∴ rank is < 3

Also

Hence rank of is A−B is 2

Characteristic equation of the given matrix is

Every matrix satisfies its own characteristic equation

∴A³ − 9A² − 47A = 20I

Applying R2 → R2 – R1

Now, expanding from a11.

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

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

Rank ≤ 3 since all entries are 1,

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

only 1 × 1 or single entries are ≠ 0

∴ rank = 1