Question

Given the matrices $$ A=\left(\begin{array}{rrrr} 2 & 3 & 1 & -4 \\ 2 & 1 & 0 & 5 \end{array}\right), \quad B=\left(\begin{array}{rr} 2 & 4 \\ 1 & -1 \\ 3 & -1 \end{array}\right), \quad C=\left(\begin{array}{rrr} 2 & 1 & 3 \\ 4 & -1 & -2 \\ -1 & 0 & 1 \end{array}\right) $$ compute or mark as meaningless all products of two of these matrices \(\left(A B, B A, A^{2}\right.\), etc. \() ;\) of three of them \(\left(A B C, A^{2} C, A^{3}, \mathrm{etc} .\right)\).

Short answer

BA and CB are defined. All other products are not defined.

Step-by-step solution

Understand the Dimensions

First, understand the dimensions of each matrix.Matrix A is 2x4, B is 3x2, and C is 3x3.

Check Product AB

Matrix A is 2x4 and Matrix B is 3x2. The inner dimensions (4 and 3) do not match, therefore AB is not defined.

Check Product BA

Matrix B is 3x2 and Matrix A is 2x4. The inner dimensions (2 and 2) match, so BA is defined and will result in a 3x4 matrix.Calculate BA: \[B \times A = \begin{pmatrix} 2 & 4 \ 1 & -1 \ 3 & -1 \times \begin{pmatrix} 2 & 3 & 1 & -4 \ 2 & 1 & 0 & 5 \end{pmatrix} = \begin{pmatrix} 12 & 14 & 2 & 12 \ 0 & 2 & 1 & -9 \ 4 & 8 & 3 & -17 \end{pmatrix}\]

Check Product A^2

Matrix A is 2x4. The matrix product A^2 would involve multiplying A by itself. Since the inner dimensions (4 and 2) do not match, A^2 is not defined.

Check Product BC

Matrix B is 3x2 and Matrix C is 3x3. The inner dimensions (2 and 3) do not match, therefore BC is not defined.

Check Product CB

Matrix C is 3x3 and Matrix B is 3x2. The inner dimensions (3 and 3) match, so CB is defined and will result in a 3x2 matrix.Calculate CB: \[C \times B = \begin{pmatrix} 2 & 1 & 3 \ 4 & -1 & -2 \ -1 & 0 & 1 \times \begin{pmatrix} 2 & 4 \ 1 & -1 \ 3 & -1 \end{pmatrix} = \begin{pmatrix} 13 & 4 \ 5 & 20 \ 1 & -5 \end{pmatrix}\]

Check Product CA

Matrix C is 3x3 and Matrix A is 2x4. The inner dimensions (3 and 2) do not match, therefore CA is not defined.

Check Product AC

Matrix A is 2x4 and Matrix C is 3x3. The inner dimensions (4 and 3) do not match, therefore AC is not defined.

Check Product A^3

Since A^2 is not defined, A^3 is also not defined.

Check Product ABC

Since the product AB is not defined, the product ABC is also not defined.

Check Product A^2C

Since A^2 is not defined, the product A^2C is also not defined.

Check Product ABC

Since both products AB and BC are not defined, the order in which these matrices are multiplied won't result in a defined product. Therefore, A must be multiplied by B prior to multiplying by C, and since this operation is invalid, ABC is not defined.

Key concepts

Matrix Dimensions

Understanding the dimensions of matrices is crucial when learning about matrix multiplication. For each matrix, the dimensions are given in the form of rows by columns; this tells you how many rows and columns the matrix has. For example:

• Matrix A has dimensions 2x4, meaning 2 rows and 4 columns.
• Matrix B has dimensions 3x2, meaning 3 rows and 2 columns.
• Matrix C has dimensions 3x3, meaning 3 rows and 3 columns.

These dimensions help determine whether two matrices can be multiplied together. The rule is that for two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. If this condition is not met, multiplication is not defined for those matrices.

Matrix Product

The matrix product, or matrix multiplication, is an operation that takes a pair of matrices and produces another matrix. However, not all pairs of matrices can be multiplied. The matrix product AB is defined if the number of columns of A equals the number of rows of B. In other words, if A is an m×n matrix and B is an n×p matrix, then their matrix product AB will be an m×p matrix.
Let's consider some examples from the given exercise:

• AB: Matrix A is 2x4 and Matrix B is 3x2. Here, the inner dimensions do not match (4 ≠ 3), so AB is not defined.
• BA: Matrix B is 3x2 and Matrix A is 2x4. Here, the inner dimensions match (2 = 2), so BA is defined and the result will be a 3x4 matrix.

When working on these multiplications, you multiply the rows of the first matrix by the columns of the second matrix and sum the products to fill each entry in the resulting matrix.

Linear Algebra

Linear algebra, a branch of mathematics, deals with vectors, vector spaces, and linear transformations. Matrices are a key component of linear algebra, and matrix multiplication is a fundamental operation within this field.
Key concepts in linear algebra include:

• Vectors: A vector is an array of numbers arranged in a specific order. They can be visualized as points or directions in space.
• Vector Spaces: These are collections of vectors that can be scaled and added together to produce another vector within the same space.
• Linear Transformations: These are functions that take a vector as input and produce another vector as output, often represented by matrix multiplication.

Matrix multiplication in linear algebra allows us to solve systems of linear equations, describe geometric transformations, and model various real-world phenomena. It provides a powerful language for expressing and solving problems that involve linear relationships.