Question

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 \(=a \cdot A \mathrm{~b}\);

Short answer

a. The result of a+b is: $$ \left[\begin{array}{rr} 3 & -3 \\ -1 & 4 \end{array}\right] $$ b. The operation a * d (transpose) is illegal. c. The result of a * d is: $$ \left[\begin{array}{rr} 2 & -2 \\ -1 & 2 \end{array}\right] $$ d. The result of a * c is: $$ \left[\begin{array}{r} 6 \\ -3 \end{array}\right] $$ e. The operation a .* c is illegal. f. The operation a/b is illegal. g. The operation a.1~b is illegal. h. The operation a * A~b is illegal.

Step-by-step solution

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. $$ \left[\begin{array}{rr} 2 & -2 \\ -1 & 2 \end{array}\right] + \left[\begin{array}{rr} 1 & -1 \\ 0 & 2 \end{array}\right] = \left[\begin{array}{rr} (2+1) & (-2-1) \\ (-1+0) & (2+2) \end{array}\right] = \left[\begin{array}{rr} 3 & -3 \\ -1 & 4 \end{array}\right] $$

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.

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. $$ \left[\begin{array}{rr} 2 & -2 \\ -1 & 2 \end{array}\right] \cdot \left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] = \left[\begin{array}{rr} (2\times1) + (-2\times0) & (2\times0) + (-2\times1) \\ (-1\times1) + (2\times0) & (-1\times0) + (2\times1) \end{array}\right] = \left[\begin{array}{rr} 2 & -2 \\ -1 & 2 \end{array}\right] $$

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. $$ \left[\begin{array}{rr} 2 & -2 \\ -1 & 2 \end{array}\right] \cdot \left[\begin{array}{r} 1 \\ -2 \end{array}\right] = \left[\begin{array}{r} (2\times1) + (-2\times-2) \\ (-1\times1) + (2\times-2) \end{array}\right] = \left[\begin{array}{r} 6 \\ -3 \end{array}\right] $$

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.

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.

g. Illegal operation

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

h. Illegal operation

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

Key concepts

Matrix Addition

Matrix addition is a basic yet essential operation in the realm of matrix operations. For two matrices to be added together, they must share the same dimensions, meaning they have the same number of rows and columns. The process involves adding corresponding elements from each matrix.

For example, if you have matrices\\[\begin{array}{rr}2 & -2 \-1 & 2\end{array}\] and \[\begin{array}{rr}1 & -1 \0 & 2\end{array}\],\\the addition results in another matrix \[\begin{array}{rr}3 & -3 \-1 & 4\end{array}\].

This is calculated by adding elements in the first row and column of both matrices, then the first row and second column, and so on. If matrices don't have the same dimensions, the operation is not possible.

Matrix Multiplication

Matrix multiplication is another fundamental operation where two matrices are combined to form a new matrix. The rule for matrix multiplication is slightly more complex than addition. It requires that the number of columns in the first matrix matches the number of rows in the second matrix. This allows the multiplication of rows by columns to produce each element of the resulting matrix.

Consider matrices \[a\] (2x2) and \[d\] (2x2, identity matrix). You multiply rows of \[a\] by columns of \[d\] to get the new matrix:\[\begin{array}{rr}2 & -2 \-1 & 2\end{array}\] \[\begin{array}{rr}1 & 0 \0 & 1\end{array}\] = \[\begin{array}{rr}2 & -2 \-1 & 2\end{array}\].

Matrix multiplication is not commutative, meaning \(A \cdot B eq B \cdot A\) in general.

Element-wise Multiplication

Element-wise multiplication, also known as the Hadamard product, involves multiplying two matrices of the same dimensions by corresponding elements. Like matrix addition, element-wise multiplication requires that the matrices be the same size.

This operation is often denoted by the symbol \(.*\). However, it was noted in the exercise that an attempt to multiply matrices \[a\] (2x2) and \[c\] (2x1) using element-wise multiplication resulted in an illegal operation due to different dimensions. In correct element-wise multiplication, every element in matrix \(A\) should be paired with exactly one element in matrix \(B\). If this condition is not met, the multiplication cannot be executed.

Matrix Dimensions

Matrix dimensions are a critical aspect in determining the legality of many matrix operations. A matrix's dimensions are usually given in the form of rows by columns, such as 2x2 or 2x1.

These dimensions dictate what operations can be performed:

• Addition requires matrices with identical dimensions.
• Matrix multiplication needs the first matrix's columns to match the second matrix's rows.
• Element-wise operations require matrices to share the same dimensions.

Understanding matrix dimensions is essential to navigate the complexities of matrix operations legally and effectively. If dimensions are not compatible, a defined mathematical operation can't proceed, which is why careful attention to matrix size is imperative.