Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. If a square matrix has an entire row of zeros, then the determinant will always be zero.

Step-by-Step Solution

Answer

The statement is true. If a square matrix has a row or column of zeros, its determinant will indeed always be zero.

Step 1: Understand the statement

The statement claims that if a square matrix has an entire row of zeros, the determinant of that matrix will always be zero.

Step 2: Recall determinant calculation

Determinant of a square matrix is usually calculated by selecting a row or a column and making use of the 'minus-plus' rule which is also referred as cofactor expansion. When there's a row or a column of zeros, the process simplifies, as multiplying anything by zero results in zero.

Step 3: Evaluate the statement

Now if we look at the statement, if any row of a square matrix is full of zeros, all the terms in that determinant calculation will be zero since they're multiplied with zero according to the rule of determinant calculation. So, it results in the determinant being zero.