Assume that \(a, b, c\), and \(a\) are defined as follows, and calculate the results of the following operations if they are legal. If an operation is itlegal, explain why it is illegal. $$ \begin{array}{ll} a=\left[\begin{array}{rr} 2 & -2 \\ -1 & 2 \end{array}\right] & b=\left[\begin{array}{rr} 1 & -1 \\ 0 & 2 \end{array}\right] \\ c=\left[\begin{array}{r} 1 \\ -2 \end{array}\right] & d=\text { eye }(2) \end{array} $$ a. result \(=a+b\); b. result \(=a * d t\) c. result \(=a \cdot d\) i d. result \(=a \cdot c\) z e. reault \(=a \cdot * c\) ? f. result \(=a \backslash b\); g. result \(=a .1 \mathrm{~b}\) : h. result \(=a \cdot A \mathrm{~b}\);

Step by step solution

01

a. Addition (a+b)

For this operation, we will be adding the matrices a and b. The addition of two matrices is possible if they have the same dimensions. Since both a and b are 2x2 matrices, this operation is legal. To add two matrices, simply add the corresponding elements. $$ \left[\begin{array}{rr} 2 & -2 \\ -1 & 2 \end{array}\right] + \left[\begin{array}{rr} 1 & -1 \\ 0 & 2 \end{array}\right] = \left[\begin{array}{rr} (2+1) & (-2-1) \\ (-1+0) & (2+2) \end{array}\right] = \left[\begin{array}{rr} 3 & -3 \\ -1 & 4 \end{array}\right] $$

02

b. a * d (transpose)

The * symbol is for element-wise multiplication, not matrix multiplication. However, this operation involves a matrix (a) and the transpose of another matrix (d). As the dimensions don't match up, the operation is illegal.

03

c. Matrix multiplication (a * d)

Here, we will perform matrix multiplication for a and d. Matrix multiplication is possible if the number of columns in the first matrix is equal to the number of rows in the second matrix. Since both a and d are 2x2 matrices, this operation is legal. $$ \left[\begin{array}{rr} 2 & -2 \\ -1 & 2 \end{array}\right] \cdot \left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] = \left[\begin{array}{rr} (2\times1) + (-2\times0) & (2\times0) + (-2\times1) \\ (-1\times1) + (2\times0) & (-1\times0) + (2\times1) \end{array}\right] = \left[\begin{array}{rr} 2 & -2 \\ -1 & 2 \end{array}\right] $$

04

d. Matrix multiplication (a * c)

Here, we will perform matrix multiplication for a and c. Matrix multiplication is possible if the number of columns in the first matrix is equal to the number of rows in the second matrix. The matrix a is 2x2 and matrix c is 2x1, so this operation is legal. $$ \left[\begin{array}{rr} 2 & -2 \\ -1 & 2 \end{array}\right] \cdot \left[\begin{array}{r} 1 \\ -2 \end{array}\right] = \left[\begin{array}{r} (2\times1) + (-2\times-2) \\ (-1\times1) + (2\times-2) \end{array}\right] = \left[\begin{array}{r} 6 \\ -3 \end{array}\right] $$

05

e. Illegal operation

The operation specified (a .* c) is not legal because element-wise multiplication requires matrices of the same dimensions. Matrix a is 2x2 and matrix c is 2x1, so their dimensions do not match, making the operation not possible.

06

f. Illegal operation

The operation specified (a/b) involves division of matrices, which is not a legal operation in matrix algebra. One can, however, try to find the inverse of a matrix and then multiply it.

07

g. Illegal operation

The operation specified (a.1~b) is not recognized in matrix algebra and should be considered illegal as- is.

08

h. Illegal operation

The operation specified (a * A~b) is not recognized or well-defined in matrix algebra and should be considered illegal.

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