Matrices — Questions & Answers

Sample questions from Matrices

1. Q1. The order of the product [[1,2,0],[3,-1,4]] x [[2,1],[0,3],[1,-2]] is:

   • A.3 x 3
   • B.2 x 2✓
   • C.2 x 3
   • D.3 x 2
   Solution

   A 2 x 3 matrix times a 3 x 2 matrix gives a matrix of order 2 x 2.

2. Q2. For which value of k does the system 2x + 3y = 7, 4x + 6y = k have infinitely many solutions?

   • A.no such k
   • B.14✓
   • C.7
   • D.0
   Solution

   The second equation is exactly twice the first when k = 14; then the two equations describe the same line and the system has infinitely many solutions. For every other k the system is inconsistent since the coefficient matrix is singular.

3. Q3. If A = [[2,1],[3,2]] and X satisfies XA = [[1,2],[3,4]], then X equals:

   • A.[[4,-3],[6,-5]]
   • B.[[2,0],[0,1]]
   • C.[[-4,-3],[6,5]]
   • D.[[-4,3],[-6,5]]✓
   Solution

   Multiply on the right by A^(-1) = [[2,-1],[-3,2]]: X = [[1,2],[3,4]]A^(-1) = [[2 - 6, -1 + 4],[6 - 12, -3 + 8]] = [[-4,3],[-6,5]]. Note X = A^(-1)[[1,2],[3,4]] would give a different matrix, so the side of multiplication matters.

4. Q4. If A = [[1,-1],[2,3]], the equation satisfied by A is:

   • A.A^2 - 4A - 5I = O
   • B.A^2 + 4A + 5I = O
   • C.A^2 - 5A + 4I = O
   • D.A^2 - 4A + 5I = O✓
   Solution

   tr A = 4 and det A = 3 + 2 = 5, so by the 2 x 2 Cayley-Hamilton relation A^2 = 4A - 5I, i.e. A^2 - 4A + 5I = O. Direct computation of A^2 = [[-1,-4],[8,7]] confirms this.

5. Q5. The matrix A = [[0,1,0],[0,0,1],[0,0,0]] satisfies A^k = O for k ≥

   • A.1
   • B.2
   • C.3✓
   • D.4
   Solution

   C is correct. A is nilpotent. A² = [[0,0,1],[0,0,0],[0,0,0]] (non-zero). A³ = [[0,0,0],[0,0,0],[0,0,0]] = O. So A^k = O for k ≥ 3. The nilpotency index is 3.

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