Question

In Exercises 39 and 40, \(A\) is an \(n \times n\) matrix. Mark each statement True or False. Justify each answer.

40.

a. The cofactor expansion of \(\det A\) down a column is equal to the cofactor expansion along a row.

b. The determinant of a triangular matrix is the sum of the entries on the main diagonal.

Short answer

1. The given statement is false.
2. The given statement is false.

Step-by-step solution

Determine whether the given statement is true or false

a)

Theorem 1states that the determinantof an \(n \times n\) matrix can be computed by cofactor expansionacross any row or down any column. Expansion across the \(i{\mathop{\rm th}\nolimits} \) row using the cofactor in \({C_{ij}} = {\left( { - 1} \right)^{i + j}}\det {A_{ij}}\)gives \(\det A = {a_{i1}}{C_{i1}} + {a_{i2}}{C_{i2}} + ... + {a_{in}}{C_{in}}\).

Cofactor expansion down the \(j{\mathop{\rm th}\nolimits} \) column is \[\det A = {a_{1j}}{C_{1j}} + {a_{2j}}{C_{2j}} + ... + {a_{nj}}{C_{nj}}\].

Thus, statement (a) is false.

Determine whether the given statement is true or false

b)

If A is a triangular matrix, then according to theorem 2,det A is the product of the entries on its main diagonal.

Thus, statement (b) is false.