Matrices & DeterminantsmediumFree

Matrices & Determinants — JEE Maths practice question

JEE Maths question with a full step-by-step solution.

Question

In a square matrix

A

of order 3, the diagonal elements

a_{ii}

are the sum of the roots of the equation

x^2 - (a+b)x + ab = 0

;

a_{i,i+1}

are the product of the roots of the same equation;

a_{i+1,i} = 1

for all valid

i

; and the remaining elements are zero. The value of

\det(A)

is:

0

(a+b)^3

a^3 - b^3

(a^2+b^2)(a+b)

correct

Solution

Step 1: Identify the elements The roots of

x^2-(a+b)x+ab=0

are

a

and

b

, so their sum is

a+b

and their product is

ab

. Therefore:

A = \begin{pmatrix}a+b & ab & 0\\ 1 & a+b & ab\\ 0 & 1 & a+b\end{pmatrix}

Step 2: Compute the determinant by expanding along

R_1

\det(A) = (a+b)\begin{vmatrix}a+b & ab\\ 1 & a+b\end{vmatrix} - ab\begin{vmatrix}1 & ab\\ 0 & a+b\end{vmatrix}

= (a+b)\big[(a+b)^2 - ab\big] - ab(a+b)

= (a+b)\big[(a+b)^2 - ab - ab\big] = (a+b)\big[(a+b)^2 - 2ab\big]

= (a+b)(a^2+b^2)

Answer: (4)

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