Problem 7 - Advanced Engineering Mathematics

Question

In Problems 7-10, determine whether the given matrices are equal. $$ \left(\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right),\left(\begin{array}{ll} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrices are not equal; they have different dimensions.

Step 1: Understand Matrix Equality

Two matrices are equal if they have the same dimensions and each corresponding element is the same. This means they must have the same number of rows and columns, and all elements in the same position must be identical.

Step 2: Compare Dimensions of Matrices

The first matrix is a 2x3 matrix (2 rows and 3 columns), while the second matrix is a 3x2 matrix (3 rows and 2 columns). Write down their dimensions for clarity: - First matrix: 2x3 - Second matrix: 3x2

Step 3: Analyze Dimension Consistency

For matrices to be equal, their dimensions must be exactly the same. Compare the dimensions from Step 2: - First matrix: 2x3 - Second matrix: 3x2 Since the dimensions are different, the matrices cannot be equal.

Key Concepts

Matrix Dimensions

When discussing matrices, understanding their dimensions is crucial. A matrix's dimensions tell us how many rows and columns it possesses. For instance, if you encounter a matrix with dimensions 2x3, it means there are 2 rows and 3 columns. Knowing the dimensions helps in identifying a matrix's structure, much like being able to recognize a grid in a spreadsheet.

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

In our example problem, the two matrices have dimensions 2x3 and 3x2 respectively. Observing these numbers helps us recognize the fundamental differences in their structure.

Matrix Comparison

Matrix comparison involves checking two matrices to see if they are equal. For matrices to be considered equal, two main conditions must be met:

• They must have the same dimensions (the same number of rows and columns).
• Each corresponding element in both matrices should be the same.

If either of these conditions is not met, the matrices cannot be equal. In our exercise, comparing dimensions was sufficient, as the matrices had different dimensions. Therefore, there's no need to compare individual elements, as the mismatch in dimensions made them unequal right from the start.

Matrices and their Properties

Matrices possess various properties that help in operations like addition, subtraction, and multiplication. One foundational property is matrix equality, which requires understanding both matrix dimensions and the values within matrices.

• Identity Matrix: This is a special square matrix with 1s on the diagonal and 0s elsewhere.
• Zero Matrix: A matrix where all elements are zero. It can have any dimensions.
• Transpose: Switching the rows and columns of a matrix creates its transpose.

The properties make matrices versatile in solving equations, performing transformations, and more. In practical scenarios, such as our exercise, recognizing equality helps in verifying solutions and understanding relationships between data sets.