Question

Question: In Exercises 21 and 22, \(A\) and \(B\) are \(n \times n\) matrices. Mark each statement True or False. Justify each answer.

1. The determinant of \(A\) is the product of the diagonal entries in \(A\).
2. An elementary row operation on \(A\) does not change the determinant.
3. \(\left( {\det A} \right)\left( {\det B} \right) = \det AB\)
4. If \(\lambda + 5\) is a factor of the characteristic polynomial of \(A\), then 5 is an eigenvalue of \(A\).

Short answer

a. The given statement is False.

b. The given statement is False.

c. The given statement is True.

d. The given statement is False.

Step-by-step solution

Find an answer for part (a)

Assume that matrix \(A\) be a \(2 \times 2\) matrix and consider matrix as \(A = \left( {\begin{array}{*{20}{c}}2&3\\3&{ - 6}\end{array}} \right)\).

Find the determinant of a matrix \(A\).

\[\begin{array}\det A = 2\left( { - 6} \right) - 3\left( 3 \right)\\ = - 12 - 9\\ = - 21\end{array}\]

The product of diagonal entries is \( - 12\) which is not equal to the determinant of a matrix \(A\), that is, \( - 12 \ne - 21\).

Therefore, the given statement is False.

Find an answer for part (b)

According to theorem 3 (properties of the determinant); When an elementary row operation is applied on a matrix \(A\) of order \(n \times n\), then the determinant of the matrix \(A\) changes, that is, when rows of the matrix are interchanged, then the sign of the determinant changes and if the row is multiplied by any scalar factor then the determinant is also multiplied by a scalar factor.

Therefore, the given statement is False.

Find an answer for part (c)

According to Theorem 3 (Property of Determinant); \(\det AB = \left( {\det A} \right)\left( {\det B} \right)\).

Therefore, the given statement is True.

Find an answer for part (d)

To obtain the eigenvalues, we equate characteristic polynomial to 0. This implies that if \(\lambda + 5\) is the factor of the characteristic polynomial, then \( - 5\) is an eigenvalue of \(A\).

Therefore, the given statement is false.