Problem 2 - Algebra and Trigonometry

Question

(a) We can multiply two matrices only if the number of ________ in the first matrix is the same as the number of _____ in the second matrix. (b) If \(A\) is a \(3 \times 3\) matrix and \(B\) is a \(4 \times 3\) matrix, which of the following matrix multiplications are possible? \(\begin{array}{llll}{\text { (i) } A B} & {\text { (ii) } B A} & {\text { (iii) } A A} & {\text { (iv) } B B}\end{array}\)

Step-by-Step Solution

Answer

Matrix multiplications possible are (ii) \(B A\) and (iii) \(A A\).

Step 1: Understand Matrix Multiplication

To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. This condition allows each element of the resulting matrix to be computed by taking the dot product of the rows and columns from the respective matrices.

Step 2: Analyze Matrix Dimensions

Determine the dimensions of matrices \(A\) and \(B\). We are given that \(A\) is a \(3 \times 3\) matrix, meaning it has 3 rows and 3 columns. Matrix \(B\) is a \(4 \times 3\) matrix, meaning it has 4 rows and 3 columns.

Step 3: Check Multiplication Possibilities for Each Pair

Evaluate each multiplication possibility given in the options:- **(i) \(A B\):** Here, \(A\) is \(3 \times 3\) and \(B\) is \(4 \times 3\). Since the number of columns in \(A\) (3) does not equal the number of rows in \(B\) (4), \(A B\) is not possible.- **(ii) \(B A\):** Here, \(B\) is \(4 \times 3\) and \(A\) is \(3 \times 3\). The number of columns in \(B\) (3) matches the number of rows in \(A\) (3), so \(B A\) is possible.- **(iii) \(A A\):** Here, both matrices are \(3 \times 3\), and the multiplication can occur since the number of columns in the first \(A\) (3) matches the number of rows in the second \(A\) (3).- **(iv) \(B B\):** Here, both matrices are \(4 \times 3\). The number of columns in the first \(B\) (3) does not match the number of rows in the second \(B\) (4), so \(B B\) is not possible.

Key Concepts

Matrix Dimensions

Matrix dimensions are an essential aspect of working with matrices, especially in operations such as multiplication. Each matrix is defined by its size, often described as "rows by columns." For example, a matrix with 3 rows and 2 columns is expressed as a \(3 \times 2\) matrix.
Understanding matrix dimensions is crucial because they determine how matrices can interact with each other in mathematical operations. When you see a matrix labeled with its dimensions, you immediately know how many entries it contains and how it is structured.

• The first number in the dimensions denotes the number of rows.
• The second number denotes the number of columns.

Knowing this helps in visualizing matrices and planning operations, such as addition, subtraction, and notably, multiplication, where alignment of dimensions is key.

Dot Product

The dot product plays a crucial role in matrix multiplication. It is a method of multiplying two equal-length sequences of numbers (vectors) from two different matrices.
During matrix multiplication, the dot product is taken between the rows of the first matrix and the columns of the second. This is what creates the individual elements of the resulting matrix.

• The dot product between two vectors involves multiplying corresponding elements and summing up these products.
• This operation is done for each row of the first matrix with each column of the second matrix, filling up the entries of the resulting matrix.

For example, if you have row vector \([a, b]\) and column vector \([xewline y]\), their dot product is computed as \(ax + by\). This computation forms the elements in the resulting matrix formed by multiplication.

Matrix Rows and Columns

Understanding the arrangement of rows and columns in a matrix helps in performing operations like matrix multiplication efficiently.
Rows are horizontal lines of elements across a matrix, while columns are vertical lines of elements. In the context of matrix multiplication, you will be looking to align columns from the first matrix with rows from the second.

• The number of elements in a row matches the number of columns in the matrix.
• Similarly, the number of elements in a column matches the number of rows in the matrix.

Visually, think of the rows in the first matrix sliding across the columns in the second matrix, aligning each element to perform the dot product. This spatial understanding helps not only in multiplying matrices but also in visualize the structural integrity of matrix setups.

Matrix Dimensions Compatibility

Matrix dimensions compatibility is a vital condition for matrix multiplication. Two matrices can be multiplied only if certain conditions regarding their dimensions are satisfied.
For multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix. This ensures that each row from the first matrix has a corresponding column in the second matrix to engage in a dot product.

• The compatibility is satisfied when the dimension format of two matrices resembles \(m \times n\) and \(n \times p\), respectively.
• In this scenario, the resulting matrix will have dimensions \(m \times p\).

By ensuring these conditions, you are prepared to multiply matrices with confidence and accuracy, avoiding errors that stem from misunderstood dimension requirements.