Definition of Matrix Multiplication

Oct 13, 2025 by magicalceylontours.com 46 views

Evaluating the Expression $4A - 3B$

In this article, we will evaluate the expression $4A - 3B$ where $A$ and $B$ are given matrices. We will first recall the definition of matrix multiplication and then apply it to evaluate the given expression.

Matrix multiplication is a fundamental operation in linear algebra that allows us to combine two matrices to form a new matrix. Given two matrices $A$ and $B$ , the product $AB$ is defined as the matrix whose entries are the dot products of the rows of $A$ with the columns of $B$ .

Definition of Matrix Multiplication

Let $A$ be an $m \times n$ matrix and $B$ be an $n \times p$ matrix. The product $AB$ is an $m \times p$ matrix whose entries are given by:

(AB)_{ij} = \sum_{k=1}^n A_{ik} B_{kj}

To evaluate the expression $4A - 3B$ , we need to first multiply the matrix $A$ by the scalar $4$ and the matrix $B$ by the scalar $3$ . Then, we can subtract the resulting matrices.

Multiplying a Matrix by a Scalar

When we multiply a matrix by a scalar, we multiply each entry of the matrix by the scalar.

Example

Let $A = \begin{pmatrix} 6 & 0 \\ -3 & 4 \\ 9 & 5 \\ 7 & 8 \end{pmatrix}$ and $k = 4$ . Then, the product $4A$ is given by:

4A = \begin{pmatrix} 4(6) & 4(0) \\ 4(-3) & 4(4) \\ 4(9) & 4(5) \\ 4(7) & 4(8) \end{pmatrix} = \begin{pmatrix} 24 & 0 \\ -12 & 16 \\ 36 & 20 \\ 28 & 32 \end{pmatrix}

Multiplying a Matrix by a Scalar

Similarly, let $B = \begin{pmatrix} 2 & 2 \\ 3 & 1 \\ 5 & -4 \\ 10 & 7 \end{pmatrix}$ and $k = 3$ . Then, the product $3B$ is given by:

3B = \begin{pmatrix} 3(2) & 3(2) \\ 3(3) & 3(1) \\ 3(5) & 3(-4) \\ 3(10) & 3(7) \end{pmatrix} = \begin{pmatrix} 6 & 6 \\ 9 & 3 \\ 15 & -12 \\ 30 & 21 \end{pmatrix}

Subtracting Matrices

Now that we have the products $4A$ and $3B$ , we can subtract the resulting matrices to evaluate the expression $4A - 3B$ .

Example

Let $A = \begin{pmatrix} 6 & 0 \\ -3 & 4 \\ 9 & 5 \\ 7 & 8 \end{pmatrix}$ , $B = \begin{pmatrix} 2 & 2 \\ 3 & 1 \\ 5 & -4 \\ 10 & 7 \end{pmatrix}$ , $4A = \begin{pmatrix} 24 & 0 \\ -12 & 16 \\ 36 & 20 \\ 28 & 32 \end{pmatrix}$ , and $3B = \begin{pmatrix} 6 & 6 \\ 9 & 3 \\ 15 & -12 \\ 30 & 21 \end{pmatrix}$ . Then, the expression $4A - 3B$ is given by:

4A - 3B = \begin{pmatrix} 24 & 0 \\ -12 & 16 \\ 36 & 20 \\ 28 & 32 \end{pmatrix} - \begin{pmatrix} 6 & 6 \\ 9 & 3 \\ 15 & -12 \\ 30 & 21 \end{pmatrix} = \begin{pmatrix} 18 & -6 \\ -21 & 13 \\ 21 & 32 \\ -2 & 11 \end{pmatrix}

In our previous article, we evaluated the expression $4A - 3B$ where $A$ and $B$ are given matrices. In this article, we will answer some frequently asked questions related to matrix multiplication and the evaluation of the expression $4A - 3B$ .

Q: What is matrix multiplication?

A: Matrix multiplication is a fundamental operation in linear algebra that allows us to combine two matrices to form a new matrix. Given two matrices $A$ and $B$ , the product $AB$ is defined as the matrix whose entries are the dot products of the rows of $A$ with the columns of $B$ .

Q: How do I multiply a matrix by a scalar?

A: When we multiply a matrix by a scalar, we multiply each entry of the matrix by the scalar. For example, if we have a matrix $A = \begin{pmatrix} 6 & 0 \\ -3 & 4 \\ 9 & 5 \\ 7 & 8 \end{pmatrix}$ and we multiply it by the scalar $4$ , the resulting matrix is $4A = \begin{pmatrix} 24 & 0 \\ -12 & 16 \\ 36 & 20 \\ 28 & 32 \end{pmatrix}$ .

Q: How do I subtract matrices?

A: To subtract matrices, we subtract corresponding entries of the matrices. For example, if we have two matrices $A = \begin{pmatrix} 6 & 0 \\ -3 & 4 \\ 9 & 5 \\ 7 & 8 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 2 \\ 3 & 1 \\ 5 & -4 \\ 10 & 7 \end{pmatrix}$ , the difference $A - B$ is given by:

A - B = \begin{pmatrix} 6 & 0 \\ -3 & 4 \\ 9 & 5 \\ 7 & 8 \end{pmatrix} - \begin{pmatrix} 2 & 2 \\ 3 & 1 \\ 5 & -4 \\ 10 & 7 \end{pmatrix} = \begin{pmatrix} 4 & -2 \\ -6 & 3 \\ 4 & 9 \\ -3 & 1 \end{pmatrix}

Q: What is the difference between matrix multiplication and scalar multiplication?

A: Matrix multiplication and scalar multiplication are two different operations. Matrix multiplication involves combining two matrices to form a new matrix, while scalar multiplication involves multiplying each entry of a matrix by a scalar.

Q: Can I multiply a matrix by a non-numeric value?

A: No, you cannot multiply a matrix by a non-numeric value. Matrix multiplication is only defined for numeric values.

Q: Can I multiply a matrix by a matrix of different dimensions?

A: No, you cannot multiply a matrix by a matrix of different dimensions. The number of columns in the first matrix must be equal to the number of rows in the second matrix.

Q: What is the result of multiplying a matrix by the zero matrix?

A: The result of multiplying a matrix by the zero matrix is the zero matrix.

Q: What is the result of multiplying a matrix by the identity matrix?

A: The result of multiplying a matrix by the identity matrix is the original matrix.

In this article, we answered some frequently asked questions related to matrix multiplication and the evaluation of the expression $4A - 3B$ . We hope that this article has been helpful in clarifying any doubts you may have had about matrix multiplication and scalar multiplication.