Matrices (JEE Advanced)

Question 1

Let A=beginpmatrix1&1&1; 0&1&1; 0&0&1endpmatrix. The sum of all nine entries of A¹⁰ is ______. Exactly one option is correct.

Accepted Answer

B. Write A=I+N with N=beginpmatrix0&1&1; 0&0&1; 0&0&0endpmatrix. Then N²=beginpmatrix0&0&1; 0&0&0; 0&0&0endpmatrix and N³=O.
Since I commutes with N, the binomial theorem applies: A¹⁰=I+10N+binom102N²=I+10N+45N².
So A¹⁰=beginpmatrix1&10&55; 0&1&10; 0&0&1endpmatrix (the (1,3) entry is 10·1+45·1=55).
Sum of entries =3+10+10+55=78.
Trap: dropping the N² term gives the (1,3) entry as 10 and a total of 33

Question 2

Let A be any real 3×3 matrix. Starting from A, the following elementary operations are performed in the order given, each on the matrix produced by the previous step: (i) R₁rightarrow R₃; (ii) R₂→ R₂−

Accepted Answer

D. Each row operation is left multiplication by the matrix obtained by applying it to I; the column operation is right multiplication by D=diag(1,2,1). Row and column operations commute, so B=E₂E₁AD, where
E₁=beginpmatrix0&0&1; 0&1&0; 1&0&0endpmatrix (swap R₁,R₃) and E₂=beginpmatrix1&0&0; −3&1&0; 0&0&1endpmatrix (R₂→ R₂−3R₁).
The operation applied later stands on the left: P=E₂E₁. Row 1 of P is row 1

Question 3

For real t, let A(t)=beginpmatrix1&t&t²/2; 0&1&t; 0&0&1endpmatrix. Which of the following statements is(are) TRUE? One or more options may be correct.

Accepted Answer

A, C. Compute A(s)A(t): the (1,2) entry is t+s, the (2,3) entry is s+t, and the (1,3) entry is (t²)/2+st+(s²)/2=((s+t)²)/2; the diagonal is 1 and the lower part is 0. Hence A(s)A(t)=A(s+t). (A) is TRUE.
(C) A(t)A(−t)=A(0)=I. TRUE.
(D) By (A), A(1)¹⁰=A(10), whose (1,3) entry is 10²/2=50≠55. FALSE. (The value 55 comes from wrongly using 1+2+…+10.)
(B) For t=1, the (1,1) entry of A(1)^TA(1) is 1, but its (

Matrices: questions with solutions

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