Problem 8 - Algebra 1

Question

Tell whether the matrices can be added. $$ \left[\begin{array}{rrr} 3 & -2 & 0 \\ -4 & 1 & -8 \end{array}\right],\left[\begin{array}{cc} -4 & 5 \\ 10 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

No, these matrices cannot be added as they do not have the same dimensions.

Step 1: Identify the Matrix Dimensions

The dimension of a matrix is defined by the number of rows and columns it contains. The first matrix has two rows and three columns, so its dimension is 2x3. The second matrix has two rows and two columns, so its dimension is 2x2.

Step 2: Compare the Dimensions

To check whether the matrices can be added, compare their dimensions. If the dimensions are the same, the matrices can be added. If not, the matrices cannot be added.

Step 3: Can the Matrices be Added?

Looking at the dimensions of the matrices, they are not the same. The first matrix is 2x3 and the second matrix is 2x2. Since these dimensions are not equal, the matrices cannot be added.

Key Concepts

Matrix Dimensions

Matrix dimensions are fundamental to understanding the structure of algebraic matrices. A matrix's dimension is represented by the number of rows and columns it consists of, typically noted as 'rows x columns'. For instance, a matrix with three rows and two columns is denoted as a 3x2 matrix.

Knowing the dimension of a matrix is essential, particularly when performing operations such as matrix addition or multiplication. Suppose we have a matrix A with dimensions 4x3 and matrix B with dimensions 4x3. These two matrices are dimensionally identical, meaning they have the same number of rows and the same number of columns. Therefore, they meet the initial criteria that allow for matrix addition.

Different dimensions can significantly affect the operations you can perform. For instance, if our matrix A were 4x3, but matrix B was 3x4, despite having the same total number of elements, their differing arrangements prevent them from being added together.

Matrix Addition Compatibility

Matrix addition compatibility is a crucial concept in the realm of algebraic matrices. For two matrices to be compatible for addition, they must share the same dimensions. In other words, each matrix must have the identical number of rows and columns as the other. If this condition is not satisfied, the operation is undefined and cannot be carried out.

In our textbook example, the first matrix is a 2x3 matrix and the second is a 2x2. Despite both having two rows, their column count differs; therefore, these matrices are incompatible for addition.

Why is Compatibility Important?

Compatibility is important because matrix addition is performed element-wise. Each element in one matrix is added to the corresponding element in the other. Without a one-to-one correspondence between each element in the matrices due to identical dimensions, addition isn't possible as there is no clear directive on how to combine the elements.

Algebraic Matrices

Algebraic matrices are a powerful tool in linear algebra used to represent and solve systems of linear equations, among other things. An algebraic matrix consists of quantities arranged in rows and columns to form a rectangular array. These quantities could be numbers, variables, or more complex expressions.

Matrices provide a way to capture the coefficients of linear equations in a structured form, allowing for streamlined calculations. Consider a set of linear equations: each equation can form a row in a matrix with the coefficients mapping to columns. An additional matrix known as a vector could hold the constants from the right-hand side of these equations.

Operations such as addition, subtraction, multiplication, and inversion can be performed on matrices, but they each have specific rules. Matrix addition and subtraction require matrices of the same dimensions, multiplication needs the first matrix's column count to match the second matrix's row count, and inversion applies only to square matrices, which have an equal number of rows and columns.