Stewart Precalc 6e Section 10.4: The Algebra of Matrices

Let \(B = \begin{pmatrix} 3 & \dfrac{1}{2} & 5 \\ 1 & -1 & 3 \end{pmatrix}\), \(C = \begin{pmatrix} 2 & -\dfrac{5}{2} & 0 \\ 0 & 2 & -3 \end{pmatrix}\), \(D = \begin{pmatrix} 7 & 3 \end{pmatrix}\), \(H = \begin{pmatrix} 3 & 1 \\ 2 & -1 \end{pmatrix}\). Carry out the operation. (a) \(3 B + 2 C\) (b) \(2 H + D\)

Let \(A = \begin{pmatrix} 2 & -5 \\ 0 & 7 \end{pmatrix}\) and \(D = \begin{pmatrix} 7 & 3 \end{pmatrix}\). Carry out the operation, or explain why it cannot be performed. (a) \(A D\)

Let \(B = \begin{pmatrix} 3 & \dfrac{1}{2} & 5 \\ 1 & -1 & 3 \end{pmatrix}\) and \(F = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\). Carry out the operation, or explain why it cannot be performed. (a) \(B^2\) (b) \(F^2\)

Let \(A = \begin{pmatrix} 2 & -5 \\ 0 & 7 \end{pmatrix}\). Carry out the operation. (a) \(A^2\) (b) \(A^3\)

Let \(A = \begin{pmatrix} 2 & -5 \\ 0 & 7 \end{pmatrix}\), \(B = \begin{pmatrix} 3 & \dfrac{1}{2} & 5 \\ 1 & -1 & 3 \end{pmatrix}\), \(D = \begin{pmatrix} 7 & 3 \end{pmatrix}\). Carry out the operation. (a) \((D A) B\) (b) \(D(A B)\)

Let \(B = \begin{pmatrix} 3 & \dfrac{1}{2} & 5 \\ 1 & -1 & 3 \end{pmatrix}\), \(C = \begin{pmatrix} 2 & -\dfrac{5}{2} & 0 \\ 0 & 2 & -3 \end{pmatrix}\), \(D = \begin{pmatrix} 7 & 3 \end{pmatrix}\), \(E = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}\), \(F = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\). Carry out the operation. (a) \(D B + D C\) (b) \(B F + F E\)

Solve for \(x\) and \(y\). \(\begin{pmatrix} x & 2 y \\ 4 & 6 \end{pmatrix} = \begin{pmatrix} 2 & -2 \\ 2 x & -6 y \end{pmatrix}\)

Solve for \(x\) and \(y\). \(3 \begin{pmatrix} x & y \\ y & x \end{pmatrix} = \begin{pmatrix} 6 & -9 \\ -9 & 6 \end{pmatrix}\)

Solve for \(x\) and \(y\). \(2 \begin{pmatrix} x & y \\ x + y & x - y \end{pmatrix} = \begin{pmatrix} 2 & -4 \\ -2 & 6 \end{pmatrix}\)

Solve for \(x\) and \(y\). \(\begin{pmatrix} x & y \\ -y & x \end{pmatrix} - \begin{pmatrix} y & x \\ x & -y \end{pmatrix} = \begin{pmatrix} 4 & -4 \\ -6 & 6 \end{pmatrix}\)

Write the system of equations as a matrix equation. \(\begin{cases} 2 x - 5 y = 7 \\ 3 x + 2 y = 4 \end{cases}\)

Write the system of equations as a matrix equation. \(\begin{cases} 6 x - y + z = 12 \\ 2 x + z = 7 \\ y - 2 z = 4 \end{cases}\)

Write the system of equations as a matrix equation. \(\begin{cases} 3 x_1 + 2 x_2 - x_3 + x_4 = 0 \\ x_1 - x_3 = 5 \\ 3 x_2 + x_3 - x_4 = 4 \end{cases}\)

Write the system of equations as a matrix equation. \(\begin{cases} x - y + z = 2 \\ 4 x - 2 y - z = 2 \\ x + y + 5 z = 2 \\ -x - y - z = 2 \end{cases}\)

Let \(A = \begin{pmatrix} 1 & 0 & 6 & -1 \\ 2 & \dfrac{1}{2} & 4 & 0 \end{pmatrix}\) (\(2 \times 4\)), \(B = \begin{pmatrix} 1 & 7 & -9 & 2 \end{pmatrix}\) (\(1 \times 4\)), \(C = \begin{pmatrix} 1 \\ 0 \\ -1 \\ -2 \end{pmatrix}\) (\(4 \times 1\)). Determine which of the products \(A B\), \(A C\), \(B A\), \(B C\), \(C A\), \(C B\) are defined, and calculate them.

(a) Prove that if \(A\) and \(B\) are \(2 \times 2\) matrices, then \((A + B)^2 = A^2 + A B + B A + B^2\). (b) If \(A\) and \(B\) are \(2 \times 2\) matrices, is it necessarily true that \((A + B)^2 = A^2 + 2 A B + B^2\)?

Jaeger Foods produces tomato sauce and tomato paste in small, medium, large, and giant cans. Can sizes (ounces): \(A = \begin{pmatrix} 6 & 10 & 14 & 28 \end{pmatrix}\). Daily production: \(B = \begin{pmatrix} 2000 & 2500 \\ 3000 & 1500 \\ 2500 & 1000 \\ 1000 & 500 \end{pmatrix}\) (rows = small, medium, large, giant; columns = sauce, paste). (a) Calculate \(A B\). (b) Interpret the entries.

Three siblings, Amy, Beth, and Chad, sell melons, squash, and tomatoes. Saturday pounds: \(A = \begin{pmatrix} 120 & 50 & 60 \\ 40 & 25 & 30 \\ 60 & 30 & 20 \end{pmatrix}\). Sunday pounds: \(B = \begin{pmatrix} 100 & 60 & 30 \\ 35 & 20 & 20 \\ 60 & 25 & 30 \end{pmatrix}\) (rows = Amy, Beth, Chad; columns = melons, squash, tomatoes). Prices per pound (dollars): \(C = \begin{pmatrix} 0.10 \\ 0.50 \\ 1.00 \end{pmatrix}\). Perform each operation and interpret. (a) \(A C\) (b) \(B C\) (c) \(A + B\) (d) \((A + B) C\)

A four-level gray scale (\(0\) = white, \(3\) = black) is used. (a) Use the gray scale to find a \(6 \times 6\) matrix that digitally represents the image in the figure. (b) Find a matrix representing a darker version. (c) Find a matrix representing the negative of the image (reverse light and dark). How do you change the elements? (d) Increase the contrast of the matrix from (b) by changing each \(1\) to \(0\) and each \(2\) to \(3\). Draw the resulting image and discuss clarity. The image matrix is \(I = \begin{pmatrix} 1 & 2 & 3 & 3 & 2 & 0 \\ 0 & 3 & 0 & 1 & 0 & 1 \\ 1 & 3 & 2 & 3 & 0 & 0 \\ 0 & 3 & 0 & 1 & 0 & 1 \\ 1 & 3 & 3 & 2 & 3 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \end{pmatrix}\).

When Are Both Products Defined? What must be true about the dimensions of \(A\) and \(B\) if both products \(A B\) and \(B A\) are defined?

Powers of a Matrix. Let \(A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\). Calculate \(A^2, A^3, A^4, ...\) until you detect a pattern. Write a general formula for \(A^n\).

Powers of a Matrix. Let \(A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}\). Calculate \(A^2, A^3, A^4, ...\) until you detect a pattern. Write a general formula for \(A^n\).

Square Roots of Matrices. A square root of a matrix \(B\) is a matrix \(A\) such that \(A^2 = B\). Find as many square roots as you can of each matrix: \(\begin{pmatrix} 4 & 0 \\ 0 & 9 \end{pmatrix}\) and \(\begin{pmatrix} 1 & 5 \\ 0 & 9 \end{pmatrix}\).

Find \(a\), \(b\), \(c\), and \(d\), if \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 5 & 2 \end{pmatrix} \)

Let \( A = \begin{pmatrix} 2 & -3 \\ 0 & 5 \\ 7 & -\dfrac{1}{2} \end{pmatrix}, B = \begin{pmatrix} 1 & 0 \\ -3 & 1 \\ 2 & 2 \end{pmatrix} \) \( C = \begin{pmatrix} 7 & -3 & 0 \\ 0 & 1 & 5 \end{pmatrix}, D = \begin{pmatrix} 6 & 0 & -6 \\ 8 & 1 & 9 \end{pmatrix} \) Carry out each indicated operation, or explain why it cannot be performed.

Solve the matrix equation \( 2 X - A = B \) for the unknown matrix \(X\), where \( A = \begin{pmatrix} 2 & 3 \\ -5 & 1 \end{pmatrix}, B = \begin{pmatrix} 4 & -1 \\ 1 & 3 \end{pmatrix} \)

Let \(A = \begin{pmatrix} 1 & 3 \\ -1 & 0 \end{pmatrix}\) and \(B = \begin{pmatrix} -1 & 5 & 2 \\ 0 & 4 & 7 \end{pmatrix}\). Calculate, if possible, the products \(A B\) and \(B A\).

Let \(A = \begin{pmatrix} 5 & 7 \\ -3 & 0 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 2 \\ 9 & -1 \end{pmatrix}\). Calculate the products \(A B\) and \(B A\).

Show that the following matrix equation is equivalent to the system of equations in Example 2 of Section 10.3: \( \begin{pmatrix} 1 & -1 & 3 \\ 1 & 2 & -2 \\ 3 & -1 & 5 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 4 \\ 10 \\ 14 \end{pmatrix} \).

In a certain city the proportions of voters in each age group who are registered as Democrats, Republicans, or Independents are given by the matrix \(A = \begin{pmatrix} 0.30 & 0.60 & 0.50 \\ 0.50 & 0.35 & 0.25 \\ 0.20 & 0.05 & 0.25 \end{pmatrix}\), where the rows correspond to Democrat, Republican, and Independent and the columns correspond to age groups 18-30, 31-50, and over 50. The distribution by age and sex of the voting population is given by \(B = \begin{pmatrix} 5000 & 6000 \\ 10000 & 12000 \\ 12000 & 15000 \end{pmatrix}\), where the rows correspond to the age groups and the columns correspond to Male and Female. Assume that within each age group, political preference is not related to gender. (a) Calculate the product \(A B\). (b) How many males are registered as Democrats in this city? (c) How many females are registered as Republicans?