Use the appropriate property of determinants from this section to justify the true statement. $$( - 5 ) \left| \begin{array} { c c c }
3 & 1 & - 1 \
7 & 3 & 1 \
3 & 1 & 7
\end{array} \right| = \left| \begin{array} { c c c }
3 & - 5 & - 1 \
7 & - 15 & 1 \
3 & - 5 & 7
\end{array} \right|$$ Do not evaluate the determinants.
A) Property number 4: If square matrix B is obtained from square matrix A by adding k times a row (or column) of A to another row (or column) of A , then | B | = | A |.
B) Property number 2: If square matrix B is obtained from square matrix A by interchanging two rows (or two columns), then | B | = $-$ | A |.
C) Property number 1: If any row (or column) of a square matrix A contains only zeros, then | A | = 0.
D) Property number 5: If two rows (or columns) of a square matrix A are identical, then | A | = 0.
E) Property number 3: If square matrix B is obtained from square matrix A by multiplying each element of any row (or column) of A by some real number k , then | B | = k | A |.

Question

Use the appropriate property of determinants from this section to justify the true statement. $$( - 5 ) \left| \begin{array} { c c c } 
3 & 1 & - 1 \
7 & 3 & 1 \
3 & 1 & 7
\end{array} \right| = \left| \begin{array} { c c c } 
3 & - 5 & - 1 \
7 & - 15 & 1 \
3 & - 5 & 7
\end{array} \right|$$ Do not evaluate the determinants.
A) Property number 4: If square matrix B is obtained from square matrix A by adding k times a row (or column) of A to another row (or column) of A , then | B | = | A |.
B) Property number 2: If square matrix B is obtained from square matrix A by interchanging two rows (or two columns), then | B | = $-$ | A |.
C) Property number 1: If any row (or column) of a square matrix A contains only zeros, then | A | = 0.
D) Property number 5: If two rows (or columns) of a square matrix A are identical, then | A | = 0.
E) Property number 3: If square matrix B is obtained from square matrix A by multiplying each element of any row (or column) of A by some real number k , then | B | = k | A |.

Quizplus · Accepted Answer

The given determinant is multiplied by (-5) in the first row. Property number 3 states that if a square matrix B is obtained from square matrix A by multiplying each element of any row (or column) of A by some real number k, then |B| = k|A|. Therefore, the given statement is true by property number 3 with k = (-5).

Use the Appropriate Property of Determinants from This Section to Justify