6.05 Matrix multiplication

For each pair of matrices, determine whether the product AB or BA is possible. If possible, determine the size of the resulting matrix.

For these matrices:

J=\left[ \begin{matrix} 1 & -2 \\ 0 & 3 \end{matrix}\right],\quad K=\left[\begin{matrix} 4 & 5 & 6 \\ -1 & 2 & 3 \end{matrix}\right],\quad L=\left[\begin{matrix} 7 \\ -8 \end{matrix} \right],\quad M=\left[\begin{matrix} 9 & 1\\ 0 & 3\\ 2 & -1 \end{matrix} \right]

Determine whether the product is possible. If possible, determine the size of the resulting matrix.

For these matrices:

A=\left[ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}\right],\quad B=\left[\begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix}\right],\quad C=\left[\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix} \right],

D=\left[\begin{matrix} 7 & 8 & 9\\ 10 & 11 & 12 \end{matrix} \right],\quad E=\left[\begin{matrix} 13 & 14 \\ 15 & 16 \\ 17 & 18 \end{matrix}\right],\quad F=\left[\begin{matrix} 19 \\ 20 \\ 21 \end{matrix}\right]

Determine whether these matrix multiplications are possible. If possible, determine the size of the resulting matrix.

For each pair of matrices A and B:

Consider the matrices A =\begin{bmatrix} 2 & 3 & 7 \\ -6 & -4 & 9 \end{bmatrix} and B = \begin{bmatrix} 0 & -3 \\ -1 & 8 \\ 6 & -2 \end{bmatrix}.

Consider the row matrix R=\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} and the column matrix C=\begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}.

Calculate:

Consider A=\begin{bmatrix} 2 & 7 \\ 3 & 8 \\ \end{bmatrix}and I=\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}.

Let E=\left[\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}\right] be the zero matrix, and F=\left[\begin{matrix} 5 & 2 \\ 7 & 3 \end{matrix}\right]

Consider A = \begin{bmatrix} 7 & 9 & -2 \\ 4 & 1 & -3 \end{bmatrix}. If A is multiplied by a column matrix, B, what is the size of matrix B?

If A is a 4 \times 2 matrix and the product AB is a 4 \times 5 matrix, what is the size of B?

A matrix calculation of A = BC resulted in the matrix A = \begin{bmatrix} 6 & -2 & 3 \\ 2 & 1 & -3 \\ -5 & 8 & -1 \end{bmatrix}.

If B is a 3 \times 2 matrix, what is the size of C?

The matrix A = \begin{bmatrix} 6 & -2 & 3 \\ 2 & 1 & -3 \\ -5 & 8 & -1 \end{bmatrix} is used in the matrix multiplication BAC, where BAC is a 3 \times 3 matrix.

Matrix F = \begin{bmatrix} 7 & 8 \end{bmatrix} is to be multiplied by matrix G to obtain a 1 \times 1 matrix.

Compare and contrast scalar multiplication with matrix multiplication in terms of what each involves, when each is possible, and the size of the resulting matrix.

Matrix P is a 2 \times 4 matrix while Q is a 4 \times 2 matrix. Explain whether these matrix expressions involving P and Q are possible and justify your answers.

Solve these equations for n:

Consider these 2\times2 matrices:

A=\begin{bmatrix} 3 & 5 \\ 3 & 2 \end{bmatrix},\, B=\begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}, \text{ and} \enspace C=\begin{bmatrix} 5 & 4 \\ 5 & 4 \end{bmatrix}

Given the matrices:

A=\begin{bmatrix} 2 & 4 & 6 \\ 0 & 1 & -1 \\ 3 & 5 & 7 \end{bmatrix},\, B=\begin{bmatrix} -1 & 1 & 0 \\ 2 & -2 & -3 \\ 4 & -4 & 5 \end{bmatrix},\, C=\begin{bmatrix} 3 & -3 & 2 \\ 1 & -1 & 0 \\ 2 & -2 & 3 \end{bmatrix}

Show that A\left(B+C\right) = AB + AC.

The transpose of a matrix A, denoted A^T, is formed by swapping the rows and columns of A. For example, if A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}, then transpose A^T is A^T = \begin{bmatrix} a & c \\ b & d \end{bmatrix}.

The 2 \times 2 matrices A, B, and C are given:

A=\begin{bmatrix} 1 & x \\ 4 & 2 \end{bmatrix},\, B=\begin{bmatrix} 3 & 2 \\ 2 & 5 \end{bmatrix},\, C=\begin{bmatrix} 2x+3 & 9 \\ 11-x & 10 \end{bmatrix}

Determine the value of x, given B^{T} A^{T} = C.

A summing matrix is a row or column matrix where all the elements are 1. The purpose of a summing matrix is to determine the total sum of either the columns or the rows.

Construct a matrix multiplication to the sum the numbers 5, 8, 3, 2, 6.

Consider these matrix equation: \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 11 \end{bmatrix}

Identify the two linear equation that this matrix equation represents. Explain how the matrix format corresponds to individual linear equation.

Determine the values of a and b. \left[\begin{matrix} 4 & 1 \\ 3 & 2 \end{matrix} \right] \times \left[\begin{matrix} a \\ 5 \end{matrix} \right] = \left[\begin{matrix} 7 & 2 \\ -1 & 4 \end{matrix} \right] \times \left[\begin{matrix} 3 \\ b \end{matrix} \right]

In linear algebra, a matrix can represent a transformation that maps points from one space to another. For example, when a matrix A is multiplied by a point P, the resulting point is the transformed point Q. This concept is useful in various fields such as computer graphics, physics, and engineering to model transformation such as rotations, translations, and scalings.

The 2 \times 2 matrix T is defined as: T=\begin{bmatrix} 3 & c \\ d & -1 \end{bmatrix} where c and d are constants.

The matrix T maps the point M\left(3,4\right) onto the point N\left(5,1\right). Determine the value of c and d.

In transportation logistics, matrices can be used to model and analyse different aspects such as speed, time, and distance for various routes. Suppose a company is analysing the transportation of goods between three cities: City A, City B, and City C. These matrices represent the speeds of vehicles in \left(\operatorname{km/h}\right) on different routes and the time taken (in hours) for these routes:

Let S be the speed matrix: S=\begin{bmatrix} 60 & 50 & 70 \\ 80 & 65 & 55 \end{bmatrix} and T be the time matrix: \begin{bmatrix} 2 & 3 \\ 1.5 & 2 \\ 2.5 & 3 \end{bmatrix}

Calculate and interpret the distance matrix D, which represents the distances covered by each vehicle on each route for two different days.