Let A=[1; 1; 1] and B=[9², -10², 11²; 12², 13², -14²; -15², 16², 17²] , then the value of A′ B A is:

Solution

Maths sir · 4 steps

Sandwiching B between a row of ones and a column of ones just adds up every entry of B.

1. Check the sizes. A' is 1 × 3, B is 3 × 3 and A is 3 × 1. So A'BA is a 1 × 1 matrix, a single number.
2. What it computes. BA multiplies each row of B by a column of ones, which gives the row sums. Then A' adds those row sums. So A'BA is the sum of all nine entries of B.

   A'BA = Σᵢ₌₁³Σⱼ₌₁³ bᵢⱼ

3. Row sums. Row 1: 81 - 100 + 121 = 102. Row 2: 144 + 169 - 196 = 117. Row 3: -225 + 256 + 289 = 320.
4. So. A'BA = 102 + 117 + 320 = 539, which is option (d).

The idea to keep. A row of ones times a matrix times a column of ones gives the total of all its entries. Spot this shortcut and skip the full matrix multiplication.

Answer from the NTA final answer key. Solution by PrepWiser.