Chapter 10, Problem 15RCC

(a)

To determine

To describe: theminor $M_{ij}$ of the element $a_{ij}$

Answer to Problem 15RCC

The minor $M_{ij}$ of the element $a_{ij}$ is discussed.

Explanation of Solution

Given:

Calculation:

Assume A is a $n\times n$ matrix; the minor $M_{ij}$ , of the element $a_{ij}$ , is defined as the determinant a $\left(n-1\right)\times \left(n-1\right)$ matrix obtained by deleting the $ith$ row and $jth$ column of A.

Conclusion:

Therefore, the minor $M_{ij}$ of the element $a_{ij}$ is discussed.

(b)

To determine

To find:the cofactor $A_{ij}$

Answer to Problem 15RCC

The cofactor $A_{ij}={\left(-1\right)}^{i+j}\cdot M_{ij}$ .

Explanation of Solution

Calculation:

The cofactor $A_{ij}$ , of an element $a_{ij}$ in a $n\times n$ matrix is defined as:

  $A_{ij}={\left(-1\right)}^{i+j}\cdot M_{ij}$

Where $M_{ij}$ is the minor of the element $a_{ij}$ .

Conclusion:

Therefore, the cofactor $A_{ij}={\left(-1\right)}^{i+j}\cdot M_{ij}$ .

(c)

To determine

To find: the determinant of A

Answer to Problem 15RCC

The procedure of finding the determinant of A is discussed.

Explanation of Solution

Calculation:

The determinant of a square $n\times n$ matrix A can be computed by the following procedure:

• Choose a row iand expand along that row by multiplying each element in the row by its corresponding cofactor $A_{ij}$ .
• If the resulting expression contains minors $M_{ij}$ that represent the determinant of a $2\times 2$ matrix, use the following formula:

  $\det \left(B\right)=b_{11}{b}_{22}-b_{12}{b}_{21}$

Where the $b_{ij}$ represent the entries of the matrix.

• Otherwise, continue to expand the matrix until you obtain $2\times 2$ determinants.

Conclusion:

Therefore,theprocedure of finding the determinant of A is discussed.

(d)

To determine

To explain: A has an inverse.

Answer to Problem 15RCC

A square $n\times n$ matrix A has an inverse $A^{-1}$ if its determinant is not equal to zero.

Explanation of Solution

Calculation:

A square $n\times n$ matrix A has an inverse $A^{-1}$ if its determinant is not equal to zero.

That is, $\det \left(A\right)\ne 0$

Conclusion:

Therefore,a square $n\times n$ matrix A has an inverse $A^{-1}$ if its determinant is not equal to zero.