Chapter 10, Problem 13RCC

(a)

To determine

To explain: the identitymatrix $I_n$ .

Answer to Problem 13RCC

The identity matrix $I_n$ is a $n\times n$ matrix.

Explanation of Solution

The identity matrix $I_n$ is a $n\times n$ matrix whose main diagonal $\left(a_{11},a_{22},\mathrm{...},a_{nn}\right)$ contains all 1s. All other entries in the matrix are zero.

(b)

To determine

To find:the inverse matrix

Answer to Problem 13RCC

If A is a $n\times n$ matrix, then its inverse $A^{-1}$ is also a $n\times n$ matrix.

Explanation of Solution

If A is a $n\times n$ matrix, then its inverse $A^{-1}$ is also a $n\times n$ matrix that satisfies the following property:

  $A{A}^{-1}=A^{-1}A=I_n$

Where

  $I_n$ is the $n\times n$ identity matrix.

(c)

To determine

To write: the formula for the inverse of a $2\times 2$ matrix.

Answer to Problem 13RCC

The formula for the inverse matrixof a $2\times 2$ matrix is $a_{11}\cdot a_{22}-a_{12}\cdot a_{21}\ne 0$ .

Explanation of Solution

Let A be a $2\times 2$ matrix defined as:

  $A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$

The inverse of A is the matrix $A^{-1}$ given by the formula:

  $A=\frac{1}{a_{11}\cdot a_{22}-a_{12}\cdot a_{21}}\cdot \left[\begin{array}{cc}{a}_{22}& -a_{12}\\ -a_{21}& a_{11}\end{array}\right]$

Observe that this formula is valid only if $a_{11}\cdot a_{22}-a_{12}\cdot a_{21}\ne 0$ .

Otherwise, A does not have an inverse matrix.

Conclusion:

Therefore,the formula for the inverse matrixof a $2\times 2$ matrix is $a_{11}\cdot a_{22}-a_{12}\cdot a_{21}\ne 0$ .

(d)

To determine

To explain: the inverse of a $3\times 3$ matrix.

Answer to Problem 13RCC

The inverse of a $3\times 3$ matrix was explained.

Explanation of Solution

Calculation:

Given a $3\times 3$ matrix A, compute its inverse by first constructing a $3\times 6$ augmented matrix of the form:

  $\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

The entries to the right of the dashed line represent the matrix A, while the entries to the left of the dashed line represent the matrix $I_3$ , the $3\times 3$ identity matrix.

To obtain the inverse, elementary row operations to transform the A matrix into reduced row-echelon form

That is

Use such operations to make the matrix on the left become the identity matrix while the matrix on the right will become the inverse $A^{-1}$ .