Calculate the following expressions (showing the details of your work) or indicate why

What properties of matrix multiplication differ from those of the multiplication of numbers? What about division of matrices?

Let A be a 50 X 50 matrix and B a 50 X 20 matrix. Are the following expressions defined or not? A + B, \(\mathbf{A}^{2}, \mathbf{B}^{2}, \mathbf{A B} . \mathbf{B A} \cdot \mathbf{A A}^{\top} . \mathbf{B}^{\top} \mathbf{A} . \mathbf{B}^{\top} \mathbf{B}, \mathbf{B} \mathbf{B}^{\top}, \mathbf{B}^{\top} \mathbf{A B}\). (Give reasons.) Text Transcription: A^2, B^2, AB, BA, AA^{top}, B^{top}A, B^{top}B, BB^{top}, B^{top}AB

How is matrix mUltiplication motivated?

Are there any linear systems without solutions? With one solution? With more than one solution? Give simple examples.

How can you give the rank of a matrix in tenns of row x + 2y + 16:;: = 3 13. 3x + 5y - 8z = ] R x + 2}' - 3z = 6 vectors? Of column vectors? Of determinants?

What is the role of rank in connection with solving linear systems?

What is the row space of a matrix? The column space? The null space?

What is the idea of Gauss elimination and back substitution?

What is the inverse of a matrix? When does it exist? How would you determine it?

What is Cramer's rule? When would you apply it?

Find all solutions or indicate that no solution exists. (Show the details of your work.) 9x - 3y = 15 5x + 4y = 48

Find all solutions or indicate that no solution exists. (Show the details of your work.) -2x - 4y + 7z = - 6 x + 2y + 16z = 3

Find all solutions or indicate that no solution exists. (Show the details of your work.) 3x + 5y - 8z = 18 x + 2y - 3z = 6

Find all solutions or indicate that no solution exists. (Show the details of your work.) 5x - 10y = 2 3x + y = 13 -x + 6y = 6

Find all solutions or indicate that no solution exists. (Show the details of your work.) -8x + 2z = 1 6y + 4z = 3 12x + 2y = 2

Find all solutions or indicate that no solution exists. (Show the details of your work.) 2y + z = -1 2x + 3y - z = -12 5x - 4y + 3z = 32

Find all solutions or indicate that no solution exists. (Show the details of your work.) 3x + 7y = 0 5x - 4y = 47 6x + 9y = 15

Find all solutions or indicate that no solution exists. (Show the details of your work.) -x + 4y - 2z = 1 3x + 4y + 6z = 1 \(x-2 y+2 z=-\frac{1}{2}\) Text Transcription: x - 2y + 2z = -1 / 2

Find all solutions or indicate that no solution exists. (Show the details of your work.) 7x + 9y - 14z = 36 -x - 3y + 2z = -12 2x + y - 4z = 4

Calculate the following expressions (showing the details of your work) or indicate why they do not exist, when \(\mathbf{A}=\left[\begin{array}{rrr}9 & 2 & 8 \\2 & 18 & 10 \\8 & 10 & 15\end{array}\right]\), \(\mathbf{B}=\left[\begin{array}{rrr}0 & 2 & 6 \\-2 & 0 & -3 \\-6 & 3 & 0\end{array}\right]\), \(\mathbf{a}=\left[\begin{array}{l}3 \\7 \\1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l}4 \\0 \\2\end{array}\right]\) AB, BA Text Transcription: A = [9 2 8 \\2 18 10 \\8 10 15] B = [0 2 6 \\ -2 0 -3 \\ -6 3 0] a= [3 \\7 \\1], b = [{4 \\0 \\2]

Calculate the following expressions (showing the details of your work) or indicate why they do not exist, when \(\mathbf{A}=\left[\begin{array}{rrr}9 & 2 & 8 \\2 & 18 & 10 \\8 & 10 & 15\end{array}\right]\), \(\mathbf{B}=\left[\begin{array}{rrr}0 & 2 & 6 \\-2 & 0 & -3 \\-6 & 3 & 0\end{array}\right]\), \(\mathbf{a}=\left[\begin{array}{l}3 \\7 \\1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l}4 \\0 \\2\end{array}\right]\) \(\mathbf{A}-\mathbf{A}^{\mathbf{T}}\) Text Transcription: A = [9 2 8 \\2 18 10 \\8 10 15] B = [0 2 6 \\ -2 0 -3 \\ -6 3 0] a= [3 \\7 \\1], b = [{4 \\0 \\2] A - A^T

Calculate the following expressions (showing the details of your work) or indicate why they do not exist, when \(\mathbf{A}=\left[\begin{array}{rrr}9 & 2 & 8 \\2 & 18 & 10 \\8 & 10 & 15\end{array}\right]\), \(\mathbf{B}=\left[\begin{array}{rrr}0 & 2 & 6 \\-2 & 0 & -3 \\-6 & 3 & 0\end{array}\right]\), \(\mathbf{a}=\left[\begin{array}{l}3 \\7 \\1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l}4 \\0 \\2\end{array}\right]\) \(\mathbf{A}^{2}+\mathrm{B}^{2}\) Text Transcription: A = [9 2 8 \\2 18 10 \\8 10 15] B = [0 2 6 \\ -2 0 -3 \\ -6 3 0] a= [3 \\7 \\1], b = [{4 \\0 \\2] A^2 + B^2

Calculate the following expressions (showing the details of your work) or indicate why they do not exist, when \(\mathbf{A}=\left[\begin{array}{rrr}9 & 2 & 8 \\2 & 18 & 10 \\8 & 10 & 15\end{array}\right]\), \(\mathbf{B}=\left[\begin{array}{rrr}0 & 2 & 6 \\-2 & 0 & -3 \\-6 & 3 & 0\end{array}\right]\), \(\mathbf{a}=\left[\begin{array}{l}3 \\7 \\1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l}4 \\0 \\2\end{array}\right]\) det A, det B, det AB Text Transcription: A = [9 2 8 \\2 18 10 \\8 10 15] B = [0 2 6 \\ -2 0 -3 \\ -6 3 0] a= [3 \\7 \\1], b = [{4 \\0 \\2]

Calculate the following expressions (showing the details of your work) or indicate why they do not exist, when \(\mathbf{A}=\left[\begin{array}{rrr}9 & 2 & 8 \\2 & 18 & 10 \\8 & 10 & 15\end{array}\right]\), \(\mathbf{B}=\left[\begin{array}{rrr}0 & 2 & 6 \\-2 & 0 & -3 \\-6 & 3 & 0\end{array}\right]\), \(\mathbf{a}=\left[\begin{array}{l}3 \\7 \\1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l}4 \\0 \\2\end{array}\right]\) \(\mathbf{A} \mathbf{A}^{T}, \mathbf{A}^{T} \mathbf{A}\) Text Transcription: A = [9 2 8 \\2 18 10 \\8 10 15] B = [0 2 6 \\ -2 0 -3 \\ -6 3 0] a= [3 \\7 \\1], b = [{4 \\0 \\2] AA^T, A^{T}A

Calculate the following expressions (showing the details of your work) or indicate why they do not exist, when \(\mathbf{A}=\left[\begin{array}{rrr}9 & 2 & 8 \\2 & 18 & 10 \\8 & 10 & 15\end{array}\right]\), \(\mathbf{B}=\left[\begin{array}{rrr}0 & 2 & 6 \\-2 & 0 & -3 \\-6 & 3 & 0\end{array}\right]\), \(\mathbf{a}=\left[\begin{array}{l}3 \\7 \\1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l}4 \\0 \\2\end{array}\right]\) \(0.2 \mathbf{B B}^T}\) Text Transcription: A = [9 2 8 \\2 18 10 \\8 10 15] B = [0 2 6 \\ -2 0 -3 \\ -6 3 0] a= [3 \\7 \\1], b = [{4 \\0 \\2] 0.2 BB^T

Calculate the following expressions (showing the details of your work) or indicate why they do not exist, when \(\mathbf{A}=\left[\begin{array}{rrr}9 & 2 & 8 \\2 & 18 & 10 \\8 & 10 & 15\end{array}\right]\), \(\mathbf{B}=\left[\begin{array}{rrr}0 & 2 & 6 \\-2 & 0 & -3 \\-6 & 3 & 0\end{array}\right]\), \(\mathbf{a}=\left[\begin{array}{l}3 \\7 \\1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l}4 \\0 \\2\end{array}\right]\) \(\mathbf{A} \mathbf{a}, \mathbf{a}^{T} \mathbf{A}, \mathbf{a}^{T} \mathbf{A} \mathbf{a}\) Text Transcription: A = [9 2 8 \\2 18 10 \\8 10 15] B = [0 2 6 \\ -2 0 -3 \\ -6 3 0] a= [3 \\7 \\1], b = [{4 \\0 \\2] Aa, a^{T}A, a^{T}Aa

Calculate the following expressions (showing the details of your work) or indicate why they do not exist, when \(\mathbf{A}=\left[\begin{array}{rrr}9 & 2 & 8 \\2 & 18 & 10 \\8 & 10 & 15\end{array}\right]\), \(\mathbf{B}=\left[\begin{array}{rrr}0 & 2 & 6 \\-2 & 0 & -3 \\-6 & 3 & 0\end{array}\right]\), \(\mathbf{a}=\left[\begin{array}{l}3 \\7 \\1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l}4 \\0 \\2\end{array}\right]\) \(\mathbf{a}^{T} \mathbf{b}, \mathbf{b}^{T} \mathbf{a}, \mathbf{a} \mathbf{b}^{T}\) Text Transcription: A = [9 2 8 \\2 18 10 \\8 10 15] B = [0 2 6 \\ -2 0 -3 \\ -6 3 0] a= [3 \\7 \\1], b = [{4 \\0 \\2] a^{T}b, b^{T}a, ab^{T}

Calculate the following expressions (showing the details of your work) or indicate why they do not exist, when \(\mathbf{A}=\left[\begin{array}{rrr}9 & 2 & 8 \\2 & 18 & 10 \\8 & 10 & 15\end{array}\right]\), \(\mathbf{B}=\left[\begin{array}{rrr}0 & 2 & 6 \\-2 & 0 & -3 \\-6 & 3 & 0\end{array}\right]\), \(\mathbf{a}=\left[\begin{array}{l}3 \\7 \\1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l}4 \\0 \\2\end{array}\right]\) \(\mathbf{b}^{T} \mathbf{B b}\) Text Transcription: A = [9 2 8 \\2 18 10 \\8 10 15] B = [0 2 6 \\ -2 0 -3 \\ -6 3 0] a= [3 \\7 \\1], b = [{4 \\0 \\2] b^{T}Bb

Calculate the following expressions (showing the details of your work) or indicate why they do not exist, when \(\mathbf{A}=\left[\begin{array}{rrr}9 & 2 & 8 \\2 & 18 & 10 \\8 & 10 & 15\end{array}\right]\), \(\mathbf{B}=\left[\begin{array}{rrr}0 & 2 & 6 \\-2 & 0 & -3 \\-6 & 3 & 0\end{array}\right]\), \(\mathbf{a}=\left[\begin{array}{l}3 \\7 \\1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l}4 \\0 \\2\end{array}\right]\) \(\mathbf{a}^{T} \mathbf{B}, \mathbf{B}^{T} \mathbf{a}\) Text Transcription: A = [9 2 8 \\2 18 10 \\8 10 15] B = [0 2 6 \\ -2 0 -3 \\ -6 3 0] a= [3 \\7 \\1], b = [{4 \\0 \\2] a^{T}B, B^{T}a

Calculate the following expressions (showing the details of your work) or indicate why they do not exist, when \(\mathbf{A}=\left[\begin{array}{rrr}9 & 2 & 8 \\2 & 18 & 10 \\8 & 10 & 15\end{array}\right]\), \(\mathbf{B}=\left[\begin{array}{rrr}0 & 2 & 6 \\-2 & 0 & -3 \\-6 & 3 & 0\end{array}\right]\), \(\mathbf{a}=\left[\begin{array}{l}3 \\7 \\1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l}4 \\0 \\2\end{array}\right]\) \(0.1\left(\mathbf{A}+\mathbf{A}^{\mathbf{T}}\right)\left(\mathbf{B}-\mathbf{B}^{\mathrm{T}}\right)\) Text Transcription: A = [9 2 8 \\2 18 10 \\8 10 15] B = [0 2 6 \\ -2 0 -3 \\ -6 3 0] a= [3 \\7 \\1], b = [{4 \\0 \\2] 0.1 (A + A^T) B - B^T)

Determine the ranks of the coefficient matrix and the augmented matrix and state how many solutions the linear system will have. In Frob. 13

Determine the ranks of the coefficient matrix and the augmented matrix and state how many solutions the linear system will have. In Frob. 12

Determine the ranks of the coefficient matrix and the augmented matrix and state how many solutions the linear system will have. In Prob. 17

Determine the ranks of the coefficient matrix and the augmented matrix and state how many solutions the linear system will have. In Prob. 14

Determine the ranks of the coefficient matrix and the augmented matrix and state how many solutions the linear system will have. In Prob. 19

Determine the ranks of the coefficient matrix and the augmented matrix and state how many solutions the linear system will have. In Prob. 18

Find the inverse or state why it does not exist. (Show details.) Of the coefficient matrix in Prob. 11

Find the inverse or state why it does not exist. (Show details.) Of the coefficient matrix in Prob. 15

Find the inverse or state why it does not exist. (Show details.) Of the coefficient matrix in Prob. 16

Find the inverse or state why it does not exist. (Show details.) Of the coefficient matrix in Prob. 18

Find the inverse or state why it does not exist. (Show details.) Of the augmented matrix in Prob. 14

Find the inverse or state why it does not exist. (Show details.) Of the diagonal matrix with entries 3, -1, 5

Find the currents in the following networks.

Find the currents in the following networks.

Find the currents in the following networks.