Problem 2 - Advanced Engineering Mathematics

Question

In Problems 1-20, fill in the blanks or answer true/false. If \(\mathbf{A}\) is a \(4 \times 7\) matrix and \(\mathbf{B}\) is a \(7 \times 3\) matrix, then the size of \(A B\) is _________

Step-by-Step Solution

Answer

The size of \(AB\) is \(4 \times 3\).

Step 1: Understand the Matrix Multiplication

To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. For matrices \(\mathbf{A}\) and \(\mathbf{B}\), this condition is satisfied as \(\mathbf{A}\) has 7 columns and \(\mathbf{B}\) has 7 rows.

Step 2: Determine the Size of the Product Matrix

When multiplying two matrices, the resulting matrix, \(\mathbf{A} \mathbf{B}\), will have the number of rows of the first matrix and the number of columns of the second matrix. Since \(\mathbf{A}\) is a \(4 \times 7\) matrix and \(\mathbf{B}\) is a \(7 \times 3\) matrix, the product matrix \(\mathbf{A} \mathbf{B}\) will have \(4\) rows and \(3\) columns.

Step 3: Write the Final Answer

The size of the matrix \(\mathbf{A} \mathbf{B}\) is \(4 \times 3\).

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when dealing with matrices. Every matrix is defined by its size, specified as 'rows by columns'. For example, if a matrix \(\mathbf{A}\) has dimensions \(4 \times 7\), it has 4 rows and 7 columns.
When you observe a matrix, think of it as a grid of numbers arranged in rows and columns. Rows are the horizontal lines of this grid, while columns are the vertical ones.
This notation helps us understand how many elements are in a matrix and organize data clearly. If you ever encounter a matrix, always look carefully at its dimensions.
Remember:

• The first number represents rows.
• The second number represents columns.

For example, if \(\mathbf{B}\) is a \(7 \times 3\) matrix, it means \(\mathbf{B}\) has 7 rows and 3 columns.

Matrix Product

In mathematics, multiplying two matrices is called finding their matrix product. However, this is not as straightforward as multiplying regular numbers. Matrix multiplication involves a sequence of arithmetic operations.
To multiply two matrices \(\mathbf{A}\) and \(\mathbf{B}\), each element of the resulting matrix is computed by taking the dot product of the corresponding row of \(\mathbf{A}\) and the column of \(\mathbf{B}\).
This means you multiply each element of a row with corresponding elements of a column and sum these products.
For a more detailed insight:

• Matrix \(\mathbf{A}\) should have the same number of columns as the number of rows in matrix \(\mathbf{B}\).
• The resulting matrix product will have dimensions based on the number of rows in \(\mathbf{A}\) and the number of columns in \(\mathbf{B}\).

Matrix multiplication allows us to solve systems of equations and transformations in various fields.

Conformability in Matrices

Conformability in matrices is an essential concept for matrix multiplication. This condition ensures that the operation can be performed without mathematical errors.
For two matrices \(\mathbf{A}\) and \(\mathbf{B}\) to be conformable for multiplication, the number of columns in \(\mathbf{A}\) must match the number of rows in \(\mathbf{B}\).
This means, before multiplying, always check that:

• The column count of the first matrix equals the row count of the second matrix.

Following this rule will ensure that you can multiply the matrices correctly and result in a valid matrix.
If matrices \(\mathbf{A}\) and \(\mathbf{B}\) do not meet this requirement, they are non-conformable, and multiplication cannot proceed. Understanding conformability helps avoid mistakes in matrix operations.