Determinants

Let A= bmatrix2&0&1\\1&1&0\\1&0&1 bmatrix, B= [B1, B2, B3 ], where B1, B2, B3 are column matrices, and AB1= bmatrix1\\0\\0 bmatrix, AB2= bmatrix2\\3\\0 bmatrix, AB3= bmatrix3\\2\\1 bmatrix If =|B| and is the sum of all the diagonal elements of B, then 3+ 3 is equal to .

Answer

28

Solution

For a column bmatrixx\ \ bmatrix, A bmatrixx\ \ bmatrix = bmatrix2x + z\ + y\ + z bmatrix, so each column of B comes from three simple equations.

AB1: 2x + z = 1, x + y = 0, x + z = 0 give x = 1, y = -1, z = -1

AB2: 2x + z = 2, x + y = 3, x + z = 0 give x = 2, y = 1, z = -2

AB3: 2x + z = 3, x + y = 2, x + z = 1 give x = 2, y = 0, z = -1

Hence B = bmatrix1&2&2\\-1&1&0\\-1&-2&-1 bmatrix

= |B| = 1(-1 - 0) - 2(1 - 0) + 2(2 + 1) = -1 - 2 + 6 = 3

and = 1 + 1 + (-1) = 1

∴ 3 + 3 = 27 + 1 = 28

Answer: 28