Let A be a 3 andtimes; 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?

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Mathematics Exam > Mathematics Questions > Let A be a 3 × 3 matrix.Let x,y,z are l...

Let A be a 3 × 3 matrix.

Let x,y,z are linearly independent 3-dimensional vectors. Suppose that we have

Then find the value of the determinant of the matrix A.

Correct answer is '0'. Can you explain this answer?

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Verified Answer

Let A be a 3 × 3 matrix.Let x,y,z are linearly independent 3-dim...

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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