Problem 29

Question

Suppose that $A$ is a $4 \times 3$ matrix, $B$ is a $1 \times 3$ matrix, $C$ is a $3 \times 1$ matrix, and $D$ is a $4 \times 3$ matrix. Which of the matrix multiplications that follow are defined? Whenever it is defined, state the size of the resulting matrix.

Step-by-Step Solution

Verified

Answer

Multiplications defined: $A \times C$ (4x1), $B \times C$ (1x1), $D \times C$ (4x1).

1Step 1: Determine Compatibility of Matrix Multiplications

When multiplying two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Let's check each multiplication one by one: - For matrix multiplication $A \times B$, $A$ is 4x3 and $B$ is 1x3. The number of columns in $A$ does not match the number of rows in $B$. Thus, $A \times B$ is **not defined**.- For matrix multiplication $A \times C$, $A$ is 4x3 and $C$ is 3x1. The number of columns in $A$ matches the number of rows in $C$. Thus, $A \times C$ is defined.- For matrix multiplication $A \times D$, both are 4x3 matrices. The number of columns in $A$ does not match the number of rows in $D$. Thus, $A \times D$ is **not defined**.- For matrix multiplication $B \times C$, $B$ is 1x3 and $C$ is 3x1. The number of columns in $B$ matches the number of rows in $C$. Thus, $B \times C$ is defined.- For matrix multiplication $C \times B$, $C$ is 3x1 and $B$ is 1x3. The number of columns in $C$ does not match the number of rows in $B$. Thus, $C \times B$ is **not defined**. - For matrix multiplication $D \times C$, $D$ is 4x3 and $C$ is 3x1. The number of columns in $D$ matches the number of rows in $C$. Thus, $D \times C$ is defined.

2Step 2: Calculate Resulting Sizes of Defined Multiplications

For the defined multiplications, the size of the resulting matrix can be determined by the number of rows in the first matrix and the number of columns in the second matrix.- For $A \times C$, the resulting matrix will be of size 4x1, taken from $A$'s rows and $C$'s columns.- For $B \times C$, the resulting matrix will be of size 1x1, taken from $B$'s rows and $C$'s columns.- For $D \times C$, the resulting matrix will be of size 4x1, taken from $D$'s rows and $C$'s columns.

Key Concepts

Matrix DimensionsCompatibility of MatricesResulting Matrix Size

Matrix Dimensions

When we talk about matrices in mathematics, their dimensions are crucial. The term "matrix dimensions" refers to the number of rows and columns in a matrix. This is always expressed in the form $m \times n$, where $m$ is the number of rows, and $n$ is the number of columns. For example, a $4 \times 3$ matrix has 4 rows and 3 columns.

These dimensions are important because they determine many properties of the matrix, including its compatibility with other matrices for multiplication. When performing operations like matrix multiplication, these dimensions help us figure out whether the multiplication is possible and what the size of the result will be.

Understanding the concept of matrix dimensions is the first step in working with matrices successfully.
Always refer to the dimensions to figure out if operations can be performed.

Compatibility of Matrices

Matrix multiplication is a fundamental operation, but it doesn't always "work" like regular multiplication. The "compatibility of matrices" is a concept that helps us determine if two matrices can be multiplied.

For matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix. If this condition is not met, the two matrices are not compatible for multiplication.

For instance, if you have a matrix $A$ of dimensions $4 \times 3$ and a matrix $B$ of dimensions $1 \times 3$, you can’t multiply $A \times B$ because 3 columns of $A$ do not match 1 row of $B$.
This step is critical before attempting any multiplication, as it saves time and avoids errors.

Resulting Matrix Size

Once you've determined that two matrices can be multiplied, the next step is to figure out the size of the resulting matrix. The "resulting matrix size" is determined by the number of rows from the first matrix and the number of columns from the second matrix.

So, if you have

a first matrix of size $4 \times 3$ and a second matrix of size $3 \times 1$, the resulting matrix after multiplication will be $4 \times 1$.
This follows the rule: the new matrix takes the number of rows from the first (4) and the number of columns from the second (1).

Remember, these dimensions tell you the shape of the product matrix and reassure you that the multiplication has been planned correctly. This understanding aids in accurately conducting matrix multiplications and analyzing their results.

Problem 28

Problem 29

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