Let A be a 3 × 3 matrix.Let x,y,z are linearly independent 3-dimensional vectors. Suppose that we haveThen find the value of the determinant of the matrix A.Correct answer is '0'. Can you explain this answer?

Question

Let A be a 3 &times; 3 matrix.Let x,y,z are linearly independent 3-dimensional vectors. Suppose that we haveThen find the value of the determinant of the matrix A.Correct answer is '0'. Can you explain this answer?

Answer

Let B be the 3 × 3 matrix whose columns are the vectors x,y,z, that is,

B = [xyz]

Then we have

(If two rows are equal, then the determinant is zero. Or you may compute the determinant by the second column cofactor expansion.)

Note that the column vectors of B are linearly independent, and hence B is nonsingular matrix. Thus the det(B)≠0.

Therefore the determinant of A must be zero.

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