Discussion Questions and Problems

Discussion Questions

M5-1 List some of the ways that matrices are used.
M5-2 Explain how to calculate the determinant of a matrix.
M5-3 What are some of the uses of determinants?
M5-4 Explain how to develop a matrix of cofactors.
M5-5 What is the adjoint of a matrix?
M5-6 How is the adjoint used in finding the inverse of a matrix?

Problems

M5-7 Find the numerical values of the following determinants.
1. $| \begin{matrix} 6 & 2 \\ - 5 & 2 \end{matrix} |$ $| \begin{matrix} 6 & 2 \\ - 5 & 2 \end{matrix} |$
2. $| \begin{matrix} 3 & 7 & - 6 \\ 1 & - 1 & 2 \\ 4 & 3 & - 2 \end{matrix} |$ $| \begin{matrix} 3 & 7 & - 6 \\ 1 & - 1 & 2 \\ 4 & 3 & - 2 \end{matrix} |$
M5-8 Use determinants to solve the following set of simultaneous equations:

$\begin{array}{l} 5 X & + & 2 Y & + & 3 Z & = & 4 \\ 2 X & + & 3 Y & + & 1 Z & = & 2 \\ 3 X & + & 1 Y & + & 2 Z & = & 3 \end{array}$ $\begin{array}{l} 5 X & + & 2 Y & + & 3 Z & = & 4 \\ 2 X & + & 3 Y & + & 1 Z & = & 2 \\ 3 X & + & 1 Y & + & 2 Z & = & 3 \end{array}$
M5-9 Use matrices to write the system of equations in Problem M5-8.
M5-10 Perform the following operations.
1. Add matrix A to matrix B.
2. Subtract matrix A from matrix B.
3. Add matrix C to matrix D.
4. Add matrix C to matrix A.
$\begin{array}{l} Matrix A & = & (\begin{array}{l} 2 & 4 & 1 \\ 3 & 8 & 7 \end{array}) & Matrix C & = & (\begin{array}{l} 3 & 6 & 9 \\ 7 & 8 & 1 \\ 9 & 2 & 4 \end{array}) \\ Matrix B & = & (\begin{array}{l} 7 & 6 & 5 \\ 0 & 1 & 2 \end{array}) & Matrix D & = & (\begin{array}{l} 5 & 1 & 6 \\ 4 & 0 & 6 \\ 3 & 1 & 5 \end{array}) \end{array}$ $\begin{array}{l} Matrix A & = & (\begin{array}{l} 2 & 4 & 1 \\ 3 & 8 & 7 \end{array}) & Matrix C & = & (\begin{array}{l} 3 & 6 & 9 \\ 7 & 8 & 1 \\ 9 & 2 & 4 \end{array}) \\ Matrix B & = & (\begin{array}{l} 7 & 6 & 5 \\ 0 & 1 & 2 \end{array}) & Matrix D & = & (\begin{array}{l} 5 & 1 & 6 \\ 4 & 0 & 6 \\ 3 & 1 & 5 \end{array}) \end{array}$
M5-11 Using the matrices in Problem M5-10, determine which of the following operations are possible. If the operation is possible, give the size of the resulting matrix.
1. $B \times C$ $B \times C$
2. $C \times B$ $C \times B$
3. $A \times B$ $A \times B$
4. $C \times D$ $C \times D$
5. $D \times C$ $D \times C$
M5-12 Perform the following matrix multiplications.
1. $Matrix C = Matrix A \times Matrix B$ $Matrix C = Matrix A \times Matrix B$
2. $Matrix G = Matrix E \times Matrix F$ $Matrix G = Matrix E \times Matrix F$
3. $Matrix T = Matrix R \times Matrix S$ $Matrix T = Matrix R \times Matrix S$
4. $Matrix Z = Matrix W \times Matrix Y$ $Matrix Z = Matrix W \times Matrix Y$
$\begin{array}{l} Matrix A & = & (\begin{array}{l} 2 \\ 1 \end{array}) & Matrix B & = & (\begin{array}{l} 3 & 4 & 5 \end{array}) \\ Matrix E & = & (\begin{array}{l} 5 & 2 & 6 & 1 \end{array}) & Matrix F & = & (\begin{array}{l} 4 \\ 3 \\ 2 \\ 0 \end{array}) \\ Matrix R & = & (\begin{array}{l} 2 & 3 \\ 1 & 4 \end{array}) & Matrix S & = & (\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array}) \\ Matrix W & = & (\begin{array}{l} 3 & 5 \\ 2 & 1 \\ 4 & 4 \end{array}) & Matrix Y & = & (\begin{array}{l} 1 & 4 & 5 & 1 \\ 2 & 3 & 6 & 5 \end{array}) \end{array}$ $\begin{array}{l} Matrix A & = & (\begin{array}{l} 2 \\ 1 \end{array}) & Matrix B & = & (\begin{array}{l} 3 & 4 & 5 \end{array}) \\ Matrix E & = & (\begin{array}{l} 5 & 2 & 6 & 1 \end{array}) & Matrix F & = & (\begin{array}{l} 4 \\ 3 \\ 2 \\ 0 \end{array}) \\ Matrix R & = & (\begin{array}{l} 2 & 3 \\ 1 & 4 \end{array}) & Matrix S & = & (\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array}) \\ Matrix W & = & (\begin{array}{l} 3 & 5 \\ 2 & 1 \\ 4 & 4 \end{array}) & Matrix Y & = & (\begin{array}{l} 1 & 4 & 5 & 1 \\ 2 & 3 & 6 & 5 \end{array}) \end{array}$
M5-13 RLB Electrical Contracting, Inc., bids on the same three jobs as Blank Plumbing (see Section M5.2). RLB must supply wiring, conduits, electrical wall fixtures, and lighting fixtures. The following are needed supplies and their costs per unit:

Demand

PROJECT WIRING (ROLLS) CONDUITS WALL FIXTURES LIGHTING FIXTURES

Dormitory 50 100 10 20

Office 70 80 20 30

Apartments 20 50 30 10

ITEM COST/UNIT ($)

Wiring 1.00

Conduits 2.00

Wall fixtures 3.00

Lighting fixtures 5.00

Use matrix multiplication to compute the cost of materials at each job site.
M5-14 Transpose matrices R and S:

$\begin{array}{l} Matrix R & = & (\begin{array}{l} 6 & 8 & 2 & 2 \\ 1 & 0 & 5 & 7 \\ 6 & 4 & 3 & 1 \\ 3 & 1 & 2 & 7 \end{array}) \\ Matrix S & = & (\begin{array}{l} 3 & 1 \\ 2 & 2 \\ 5 & 4 \end{array}) \end{array}$ $\begin{array}{l} Matrix R & = & (\begin{array}{l} 6 & 8 & 2 & 2 \\ 1 & 0 & 5 & 7 \\ 6 & 4 & 3 & 1 \\ 3 & 1 & 2 & 7 \end{array}) \\ Matrix S & = & (\begin{array}{l} 3 & 1 \\ 2 & 2 \\ 5 & 4 \end{array}) \end{array}$
M5-15 Find the matrix of cofactors and adjoint of the following matrix:

$(\begin{matrix} 1 & 4 & 7 \\ 2 & 0 & 8 \\ 3 & 6 & 9 \end{matrix})$ $(\begin{matrix} 1 & 4 & 7 \\ 2 & 0 & 8 \\ 3 & 6 & 9 \end{matrix})$
M5-16 Find the inverse of the original matrix of Problem M5-15 and verify its correctness.
M5-17 Write the following as a system of equations:

$(\begin{matrix} 5 & 1 \\ 4 & 2 \end{matrix}) (\begin{matrix} X_{1} \\ X_{2} \end{matrix}) = (\begin{matrix} 240 \\ 320 \end{matrix})$ $(\begin{matrix} 5 & 1 \\ 4 & 2 \end{matrix}) (\begin{matrix} X_{1} \\ X_{2} \end{matrix}) = (\begin{matrix} 240 \\ 320 \end{matrix})$
M5-18 Write the following as a system of equations:

$(\begin{matrix} 0 & 4 & 3 \\ 1 & 2 & 2 \end{matrix}) (\begin{matrix} X_{1} \\ X_{2} \\ X_{3} \end{matrix}) = (\begin{matrix} 28 \\ 16 \end{matrix})$ $(\begin{matrix} 0 & 4 & 3 \\ 1 & 2 & 2 \end{matrix}) (\begin{matrix} X_{1} \\ X_{2} \\ X_{3} \end{matrix}) = (\begin{matrix} 28 \\ 16 \end{matrix})$

Demand
Dormitory	50	100	10	20
Office	70	80	20	30
Apartments	20	50	30	10

ITEM	COST/UNIT ($)
Wiring	1.00
Conduits	2.00
Wall fixtures	3.00
Lighting fixtures	5.00

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Demand
PROJECT	WIRING (ROLLS)	CONDUITS	WALL FIXTURES	LIGHTING FIXTURES
Dormitory	50	100	10	20
Office	70	80	20	30
Apartments	20	50	30	10

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Table of Contents for
Discussion Questions and Problems