Question

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

39.

a. An \(n \times n\) determinant is defined by determinants of \(\left( {n - 1} \right) \times \left( {n - 1} \right)\) submatrices.

b. The \(\left( {i,j} \right)\)-cofactor of a matrix \(A\) is the matrix \({A_{ij}}\) obtained by deleting from A its \(i{\mathop{\rm th}\nolimits} \) row and \[j{\mathop{\rm th}\nolimits} \]column.

Short answer

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

Step-by-step solution

Determine whether the given statement is true or false

a)

An \(n \times n\) determinant is defined by adeterminant of \(n \times n\)submatrices.

Thus, statement (a) is true.

Determine whether the given statement is true or false

b)

The \(\left( {i,j} \right)\)cofactor of \(A\) is the number \({C_{ij}}\) given by \({C_{ij}} = {\left( { - 1} \right)^{i + j}}\det {A_{ij}}\).

Thus, statement (b) is false.