Question

Let matrices \(A, B, C\), and \(D\) be of orders \(2 \times 3,2 \times 3,3 \times 2\), and \(2 \times 2\), respectively. Determine whether the matrices are of proper order to perform the operation(s). If so, give the order of the answer. $$ B C-D $$

Short answer

Yes, matrices B, C, and D can be operated as BC - D. The final order of the matrix after applying the operation BC - D would be \(2 \times 2\).

Step-by-step solution

Evaluate the Order for Multiplication

The order of matrix B is \(2 \times 3\) and the order of matrix C is \(3 \times 2\). For the multiplication of two matrices, the number of columns in the first matrix should be equal to the number of rows in the second matrix. Correspondingly, here, matrix B has 3 columns and matrix C has 3 rows, which makes them compatible for multiplication. The order of the resulting matrix will be determined by the number of rows from the first matrix (B) and the number of columns from the second matrix (C), hence, \(2 \times 2\).

Evaluate the Order for Subtraction

Now, the order of matrix D is \(2 \times 2\) which is the same as the resulting matrix of BC, \(2 \times 2\). For subtracting two matrices, both matrices should be of the same order. Therefore, matrix D can be subtracted from the matrix BC.

Provide the Final Order

As the order of matrices for multiplication and subtraction matches, it is feasible to perform the operation BC - D. The order of the final answer matrix will also be \(2 \times 2\) as the operation does not change the matrix dimensions.

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, where two matrices are combined to produce another matrix. However, not all matrices can be multiplied together; the number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible.

When matrices meet this criterion, they are said to be 'conformable' for multiplication. For instance, if matrix A is of order m × n and matrix B is of order n × p, the resulting matrix will be of order m × p. In our textbook exercise, matrix B (2 × 3) can be multiplied by matrix C (3 × 2) because the number of columns in B is equal to the number of rows in C. The product, therefore, results in a new matrix of order 2 × 2, where the first dimension is the number of rows from B, and the second dimension is the number of columns from C.

It's important to understand the individual elements in the resulting matrix are calculated by adding the products of corresponding elements from the rows of the first matrix and the columns of the second matrix. This is why the elements of the product matrix will be sums of products and not just simple multiplications of elements as in element-wise multiplication.

Matrix Order

The 'order' of a matrix, simply put, is a term used to describe its dimensions. It's given in the format of 'rows × columns'. Understanding the order of a matrix is crucial because it determines the matrix's shape and the kind of operations we can perform on it.

For example, in our textbook problem, we see matrix orders such as 2 × 3 and 3 × 2. The first number always refers to the number of rows, and the second number refers to the columns. So a 2 × 3 matrix has 2 rows and 3 columns. When performing matrix operations, the order often dictates whether the operation can proceed; it's like making sure puzzle pieces fit together before attempting to connect them.

• In matrix multiplication, the inner dimensions must match.
• In matrix addition and subtraction, the orders must be exactly the same.
• The resulting matrix after an operation takes its order from the outer dimensions of the matrices involved in multiplication or keeps the same order for addition or subtraction.

This concept is reflected in our exercise, as it shows that we have to consider the orders of matrices when attempting to perform operations like multiplication and subtraction.

Matrix Subtraction

Matrix subtraction is similar to matrix addition; it can only be performed when the matrices involved have the exact same order. In other words, they must have the same number of rows and columns.

When we subtract two matrices, we subtract their corresponding elements, which means the element in the first row and first column of the first matrix is subtracted from the element in the first row and first column of the second matrix, and so on for the rest of the elements. Given the nature of this operation, it's clear why both matrices must have the same order. In our exercise, after multiplying matrices B and C, we get a 2 × 2 matrix which can then be subtracted from matrix D, which is also 2 × 2.

It's important to keep in mind that matrix subtraction, just like addition, is an element-wise operation, and it does not change the dimension of the involved matrices, hence preserving the order.