Linear Algebra Ch 1.4 — The Matrix Equation Ax = b

1 The Matrix Equation Ax = b · Level 1

Compute the products using (a) the definition, as in Example 1, and using (b) the row–vector rule for computing \(A \mathbf{x}\). If a product is undefined, explain why. \(\begin{pmatrix} -4 & 2 \\ 1 & 6 \\ 0 & 1 \end{pmatrix} \vec{3, 1}\)

2 The Matrix Equation Ax = b · Level 1

Compute the products using (a) the definition, as in Example 1, and using (b) the row–vector rule for computing \(A \mathbf{x}\). If a product is undefined, explain why. \(\begin{pmatrix} 2 \\ 6 \\ -1 \end{pmatrix} \vec{1, -1}\)

3 The Matrix Equation Ax = b · Level 1

Compute the products using (a) the definition, as in Example 1, and using (b) the row–vector rule for computing \(A \mathbf{x}\). If a product is undefined, explain why. \(\begin{pmatrix} 6 & 5 \\ -4 & -3 \\ 7 & 6 \end{pmatrix} \vec{1, -3}\)

4 The Matrix Equation Ax = b · Level 1

Compute the products using (a) the definition, as in Example 1, and using (b) the row–vector rule for computing \(A \mathbf{x}\). If a product is undefined, explain why. \(\begin{pmatrix} 8 & 3 & -4 \\ 5 & 1 & 2 \end{pmatrix} \vec{1, 1, 1}\)

5 The Matrix Equation Ax = b · Level 1

In Exercises 5 and 6, write the system first as a vector equation and then as a matrix equation. \(\begin{pmatrix} 7 & 2 & -9 & 3 \\ -4 & -5 & 7 & -2 \end{pmatrix} \vec{6, -9, 1, -8} = \vec{-9, 44}\) Write the above matrix equation as a vector equation.

6 The Matrix Equation Ax = b · Level 1

In Exercises 5 and 6, write the system first as a vector equation and then as a matrix equation. \(\begin{pmatrix} 7 & -3 \\ 2 & 1 \\ 9 & -6 \\ -3 & 2 \end{pmatrix} \vec{-2, -5} = \vec{1, -9, 12, -4}\) Write the above matrix equation as a vector equation.

7 The Matrix Equation Ax = b · Level 1

In Exercises 7 and 8, write the vector equation as a matrix equation. \(x_1 \vec{4, -1, 7, -4} + x_2 \vec{-5, 3, -8, 0} + x_3 \vec{7, -5, 0, 2} = \vec{6, -8, 0, -7}\)

8 The Matrix Equation Ax = b · Level 1

In Exercises 7 and 8, write the vector equation as a matrix equation. \(z_1 \vec{4, -2} + z_2 \vec{-4, 5} + z_3 \vec{-5, 4} + z_4 \vec{3, 0} = \vec{4, 13}\)

9 The Matrix Equation Ax = b · Level 1

In Exercises 9 and 10, write the system first as a vector equation and then as a matrix equation. \(4 x_1 + x_2 - 7 x_3 = 8\) \(x_2 + 6 x_3 = 0\)

10 The Matrix Equation Ax = b · Level 1

In Exercises 9 and 10, write the system first as a vector equation and then as a matrix equation. \(8 x_1 - x_2 = 4\) \(5 x_1 + 4 x_2 = 1\) \(x_1 - 3 x_2 = 2\)

11 The Matrix Equation Ax = b · Level 2

Given \(A\) and \(\mathbf{b}\), write the augmented matrix for the linear system that corresponds to the matrix equation \(A \mathbf{x} = \mathbf{b}\). Then solve the system and write the solution as a vector. \(A = \begin{pmatrix} 1 & 2 & 4 \\ 0 & 1 & 5 \\ -2 & -4 & -3 \end{pmatrix}\), \(\mathbf{b} = \vec{-2, 2, 9}\)

12 The Matrix Equation Ax = b · Level 2

Given \(A\) and \(\mathbf{b}\), write the augmented matrix for the linear system that corresponds to the matrix equation \(A \mathbf{x} = \mathbf{b}\). Then solve the system and write the solution as a vector. \(A = \begin{pmatrix} 1 & 2 & 1 \\ -3 & -1 & 2 \\ 0 & 5 & 3 \end{pmatrix}\), \(\mathbf{b} = \vec{0, 1, -1}\)

13 The Matrix Equation Ax = b · Level 2

Let \(\mathbf{u} = \vec{0, 4, 4}\) and \(A = \begin{pmatrix} -3 & -5 \\ -2 & 6 \\ 1 & 1 \end{pmatrix}\). Is \(\mathbf{u}\) in the plane in \(RR^3\) spanned by the columns of \(A\)? (See the figure.) Why or why not?

14 The Matrix Equation Ax = b · Level 2

Let \(\mathbf{u} = \vec{2, -3, 2}\) and \(A = \begin{pmatrix} 5 & 8 & 7 \\ 0 & 1 & -1 \\ 1 & 3 & 0 \end{pmatrix}\). Is \(\mathbf{u}\) in the subset of \(RR^3\) spanned by the columns of \(A\)? Why or why not?

15 The Matrix Equation Ax = b · Level 3

Let \(A = \begin{pmatrix} 3 & -4 \\ -6 & 8 \end{pmatrix}\) and \(\mathbf{b} = \vec{b_1, b_2}\). Show that the equation \(A \mathbf{x} = \mathbf{b}\) does not have a solution for all possible \(\mathbf{b}\), and describe the set of all \(\mathbf{b}\) for which \(A \mathbf{x} = \mathbf{b}\) does have a solution.

16 The Matrix Equation Ax = b · Level 3

Repeat Exercise 15: \(A = \begin{pmatrix} 1 & -3 & -4 \\ -3 & 2 & 6 \\ 5 & -1 & -8 \end{pmatrix}\), \(\mathbf{b} = \vec{b_1, b_2, b_3}\) Show that \(A \mathbf{x} = \mathbf{b}\) does not have a solution for all possible \(\mathbf{b}\), and describe the set of all \(\mathbf{b}\) for which \(A \mathbf{x} = \mathbf{b}\) does have a solution.

17 The Matrix Equation Ax = b · Level 2

Exercises 17–20 refer to the matrices \(A\) and \(B\) below. Make appropriate calculations that justify your answers and mention an appropriate theorem. \(A = \begin{pmatrix} 1 & 3 & 0 & 3 \\ -1 & -1 & -1 & 1 \\ 0 & -4 & 2 & -8 \\ 2 & 0 & 3 & -1 \end{pmatrix}\), \(B = \begin{pmatrix} 1 & 3 & -2 & 2 \\ 0 & 1 & 1 & -5 \\ 1 & 2 & -3 & 7 \\ -2 & -8 & 2 & -1 \end{pmatrix}\) How many rows of \(A\) contain a pivot position? Does the equation \(A \mathbf{x} = \mathbf{b}\) have a solution for each \(\mathbf{b}\) in \(RR^4\)?

18 The Matrix Equation Ax = b · Level 2

Exercises 17–20 refer to the matrices \(A\) and \(B\) below. \(A = \begin{pmatrix} 1 & 3 & 0 & 3 \\ -1 & -1 & -1 & 1 \\ 0 & -4 & 2 & -8 \\ 2 & 0 & 3 & -1 \end{pmatrix}\), \(B = \begin{pmatrix} 1 & 3 & -2 & 2 \\ 0 & 1 & 1 & -5 \\ 1 & 2 & -3 & 7 \\ -2 & -8 & 2 & -1 \end{pmatrix}\) Do the columns of \(B\) span \(RR^4\)? Does the equation \(B \mathbf{x} = \mathbf{y}\) have a solution for each \(\mathbf{y}\) in \(RR^4\)?

19 The Matrix Equation Ax = b · Level 2

Exercises 17–20 refer to the matrices \(A\) and \(B\) below. \(A = \begin{pmatrix} 1 & 3 & 0 & 3 \\ -1 & -1 & -1 & 1 \\ 0 & -4 & 2 & -8 \\ 2 & 0 & 3 & -1 \end{pmatrix}\), \(B = \begin{pmatrix} 1 & 3 & -2 & 2 \\ 0 & 1 & 1 & -5 \\ 1 & 2 & -3 & 7 \\ -2 & -8 & 2 & -1 \end{pmatrix}\) Can each vector in \(RR^4\) be written as a linear combination of the columns of the matrix \(A\) above? Do the columns of \(A\) span \(RR^4\)?

20 The Matrix Equation Ax = b · Level 2

Exercises 17–20 refer to the matrices \(A\) and \(B\) below. \(A = \begin{pmatrix} 1 & 3 & 0 & 3 \\ -1 & -1 & -1 & 1 \\ 0 & -4 & 2 & -8 \\ 2 & 0 & 3 & -1 \end{pmatrix}\), \(B = \begin{pmatrix} 1 & 3 & -2 & 2 \\ 0 & 1 & 1 & -5 \\ 1 & 2 & -3 & 7 \\ -2 & -8 & 2 & -1 \end{pmatrix}\) Can every vector in \(RR^4\) be written as a linear combination of the columns of the matrix \(B\) above? Do the columns of \(B\) span \(RR^3\)?

21 The Matrix Equation Ax = b · Level 2

Let \(\mathbf{v}_1 = \vec{1, 0, -1, 0}\), \(\mathbf{v}_2 = \vec{0, -1, 0, 1}\), \(\mathbf{v}_3 = \vec{1, 0, 0, -1}\). Does \({\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3}\) span \(RR^4\)? Why or why not?

22 The Matrix Equation Ax = b · Level 2

Let \(\mathbf{v}_1 = \vec{0, 0, -2}\), \(\mathbf{v}_2 = \vec{0, -3, 8}\), \(\mathbf{v}_3 = \vec{4, -1, -5}\). Does \({\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3}\) span \(RR^3\)? Why or why not?

23 The Matrix Equation Ax = b · Level 2

Mark each statement True or False. Justify each answer. The equation \(A \mathbf{x} = \mathbf{b}\) is referred to as a vector equation.

24 The Matrix Equation Ax = b · Level 2

Mark each statement True or False. Justify each answer. Every matrix equation \(A \mathbf{x} = \mathbf{b}\) corresponds to a vector equation with the same solution set.

25 The Matrix Equation Ax = b · Level 2

Mark each statement True or False. Justify each answer. If the equation \(A \mathbf{x} = \mathbf{b}\) is inconsistent, then \(\mathbf{b}\) is not in the set spanned by the columns of \(A\).

26 The Matrix Equation Ax = b · Level 2

Mark each statement True or False. Justify each answer. A vector \(\mathbf{b}\) is a linear combination of the columns of a matrix \(A\) if and only if the equation \(A \mathbf{x} = \mathbf{b}\) has at least one solution.

27 The Matrix Equation Ax = b · Level 2

Mark each statement True or False. Justify each answer. The equation \(A \mathbf{x} = \mathbf{b}\) is consistent if the augmented matrix \([A \med \mathbf{b}]\) has a pivot in every row.

28 The Matrix Equation Ax = b · Level 2

Mark each statement True or False. Justify each answer. If \(A\) is an \(m \times n\) matrix whose columns do not span \(RR^m\), then the equation \(A \mathbf{x} = \mathbf{b}\) is inconsistent for some \(\mathbf{b}\) in \(RR^m\).

29 The Matrix Equation Ax = b · Level 2

Mark each statement True or False. Justify each answer. The first entry in the product \(A \mathbf{x}\) is a sum of products.

30 The Matrix Equation Ax = b · Level 2

Mark each statement True or False. Justify each answer. Any linear combination of vectors can always be written in the form \(A \mathbf{x}\) for a suitable matrix \(A\) and vector \(\mathbf{x}\).

31 The Matrix Equation Ax = b · Level 2

Mark each statement True or False. Justify each answer. If the columns of an \(m \times n\) matrix \(A\) span \(RR^m\), then the equation \(A \mathbf{x} = \mathbf{b}\) is consistent for each \(\mathbf{b}\) in \(RR^m\).

32 The Matrix Equation Ax = b · Level 2

Mark each statement True or False. Justify each answer. The solution set of a linear system whose augmented matrix is \([\mathbf{a}_1 \med \mathbf{a}_2 \med \mathbf{a}_3 \med \mathbf{b}]\) is the same as the solution set of \(A \mathbf{x} = \mathbf{b}\), if \(A = [\mathbf{a}_1 \med \mathbf{a}_2 \med \mathbf{a}_3]\).

33 The Matrix Equation Ax = b · Level 2

Mark each statement True or False. Justify each answer. If \(A\) is an \(m \times n\) matrix and if the equation \(A \mathbf{x} = \mathbf{b}\) is inconsistent for some \(\mathbf{b}\) in \(RR^m\), then \(A\) cannot have a pivot position in every row.

34 The Matrix Equation Ax = b · Level 2

Mark each statement True or False. Justify each answer. If the augmented matrix \([A \med \mathbf{b}]\) has a pivot position in every row, then the equation \(A \mathbf{x} = \mathbf{b}\) is inconsistent.

35 The Matrix Equation Ax = b · Level 3

Note that \(\begin{pmatrix} 3 & -4 & 2 \\ 6 & -3 & 4 \\ -8 & 9 & -5 \end{pmatrix} \vec{-4, -1, 3} = \vec{-2, -9, 8}\). Use this fact (and no row operations) to find scalars \(c_1, c_2, c_3\) such that \(\vec{-2, -9, 8} = c_1 \vec{3, 6, -8} + c_2 \vec{-4, -3, 9} + c_3 \vec{2, 4, -5}\).

36 The Matrix Equation Ax = b · Level 3

Let \(\mathbf{u} = \vec{7, 2, 5}\), \(\mathbf{v} = \vec{3, 1, 3}\), \(\mathbf{w} = \vec{6, 1, 0}\). It can be verified that \(3 \mathbf{u} - 5 \mathbf{v} - \mathbf{w} = \mathbf{0}\). Use this fact (and no row operations) to find \(x_1\) and \(x_2\) that satisfy the equation \(\begin{pmatrix} 7 & 3 \\ 2 & 1 \\ 5 & 3 \end{pmatrix} \vec{x_1, x_2} = \vec{6, 1, 0}\).

37 The Matrix Equation Ax = b · Level 2

Rewrite the (vector) equation \(x_1 \mathbf{q}_1 + x_2 \mathbf{q}_2 + x_3 \mathbf{q}_3 = \mathbf{v}\) as a matrix equation. (Use the vectors \(\mathbf{q}_1, \mathbf{q}_2, \mathbf{q}_3, \mathbf{v}\) in your answer.)

38 The Matrix Equation Ax = b · Level 2

Use Theorem 3 (and not row operations) to show that the equation below is true by showing that the left side equals \(A \mathbf{x}\) for appropriate \(A\) and \(\mathbf{x}\). \((-3) \vec{-3, 5} + 2 \vec{5, 8} + 4 \vec{1, -2} + (-1) \vec{-4, 9} + 2 \vec{7, -4} = \vec{-8, -1}\)

39 The Matrix Equation Ax = b · Level 3

Construct a \(3 \times 3\) matrix, not in echelon form, whose columns span \(RR^3\).

40 The Matrix Equation Ax = b · Level 3

Construct a \(3 \times 3\) matrix, not in echelon form, whose columns do NOT span \(RR^3\).

41 The Matrix Equation Ax = b · Level 2

Let \(A\) be a \(3 \times 3\) matrix with three pivot positions. Does the equation \(A \mathbf{x} = \mathbf{0}\) have a nontrivial solution? Does the equation \(A \mathbf{x} = \mathbf{b}\) have at least one solution for every \(\mathbf{b}\) in \(RR^3\)?

42 The Matrix Equation Ax = b · Level 2

Let \(A\) be a \(3 \times 3\) matrix with two pivot positions. Does the equation \(A \mathbf{x} = \mathbf{0}\) have a nontrivial solution? Does the equation \(A \mathbf{x} = \mathbf{b}\) have at least one solution for every \(\mathbf{b}\) in \(RR^3\)?

43 The Matrix Equation Ax = b · Level 2

Let \(A\) be a \(3 \times 2\) matrix with two pivot positions. Does the equation \(A \mathbf{x} = \mathbf{0}\) have a nontrivial solution? Does the equation \(A \mathbf{x} = \mathbf{b}\) have at least one solution for every \(\mathbf{b}\) in \(RR^3\)?

44 The Matrix Equation Ax = b · Level 2

Let \(A\) be a \(2 \times 4\) matrix with two pivot positions. Does the equation \(A \mathbf{x} = \mathbf{0}\) have a nontrivial solution? Does the equation \(A \mathbf{x} = \mathbf{b}\) have at least one solution for every \(\mathbf{b}\) in \(RR^2\)?

45 The Matrix Equation Ax = b · Level 2

In Exercises 45 and 46, find a nontrivial solution of \(A \mathbf{x} = \mathbf{0}\) by inspection. [Hint: Think of the equation \(A \mathbf{x} = \mathbf{0}\) as a vector equation.] \(A = \begin{pmatrix} -2 & -6 \\ 7 & 21 \\ -3 & -9 \end{pmatrix}\)

46 The Matrix Equation Ax = b · Level 2

In Exercises 45 and 46, find a nontrivial solution of \(A \mathbf{x} = \mathbf{0}\) by inspection. \(A = \begin{pmatrix} 4 & -6 \\ -8 & 12 \\ 6 & -9 \end{pmatrix}\)

47 The Matrix Equation Ax = b · Level 3

Construct a \(3 \times 3\) nonzero matrix \(A\) such that the vector \(\vec{1, 1, 1}\) is a solution of \(A \mathbf{x} = \mathbf{0}\).

48 The Matrix Equation Ax = b · Level 3

Construct a \(3 \times 3\) nonzero matrix \(A\) such that the vector \(\vec{1, -2, 1}\) is a solution of \(A \mathbf{x} = \mathbf{0}\).

49 The Matrix Equation Ax = b · Level 3

Construct a \(2 \times 2\) matrix \(A\) such that the solution set of \(A \mathbf{x} = \mathbf{0}\) is the line in \(RR^2\) through \(\vec{4, 1}\) and the origin. Then, find a vector \(\mathbf{b}\) in \(RR^2\) such that the solution set of \(A \mathbf{x} = \mathbf{b}\) is NOT a line through the origin (or, if that is not possible, explain why).

50 The Matrix Equation Ax = b · Level 3

Suppose \(A\) is a \(3 \times 3\) matrix and \(\mathbf{y}\) is a vector in \(RR^3\) such that the equation \(A \mathbf{x} = \mathbf{y}\) does not have a solution. Does there exist a vector \(\mathbf{z}\) in \(RR^3\) such that the equation \(A \mathbf{x} = \mathbf{z}\) has a unique solution? Discuss.

51 The Matrix Equation Ax = b · Level 3

Let \(A\) be an \(m \times n\) matrix, and let \(\mathbf{u}\) be a vector in \(RR^n\) that satisfies the equation \(A \mathbf{x} = \mathbf{0}\). Show that for any scalar \(c\), the vector \(c \mathbf{u}\) also satisfies \(A \mathbf{x} = \mathbf{0}\). [That is, show that \(A(c \mathbf{u}) = \mathbf{0}\).]

52 The Matrix Equation Ax = b · Level 3

Let \(A\) be an \(m \times n\) matrix, and let \(\mathbf{u}\) and \(\mathbf{v}\) be vectors in \(RR^n\) with the property that \(A \mathbf{u} = \mathbf{0}\) and \(A \mathbf{v} = \mathbf{0}\). Explain why \(A(\mathbf{u} + \mathbf{v}) = \mathbf{0}\). What can be said about \(A(c \mathbf{u} + d \mathbf{v})\) for arbitrary scalars \(c\) and \(d\)?

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