Problem 47 - Algebra and Trigonometry

Question

Let $$ \begin{array}{l}{A=\left[\begin{array}{rrrr}{1} & {0} & {6} & {-1} \\ {2} & {\frac{1}{2}} & {4} & {0}\end{array}\right]} \\\ {B=\left[\begin{array}{llll}{1} & {7} & {-9} & {2}\end{array}\right]} \\ {C} & {=\left[\begin{array}{r}{1} \\ {0} \\ {-1} \\\ {-2}\end{array}\right]}\end{array} $$ Determine which of the following products are defined, and calculate the ones that are: $$ \begin{array}{ll}{A B C} & {A C B} & {B A C} \\ {B C A} & {C A B} & {C B A}\end{array} $$

Step-by-Step Solution

Answer

None of the products are defined.

Step 1: Matrix A Dimensions

Matrix \(A\) has the dimensions \(2 \times 4\), meaning it has 2 rows and 4 columns.

Step 2: Matrix B Dimensions

Matrix \(B\) has the dimensions \(1 \times 4\), meaning it has 1 row and 4 columns.

Step 3: Matrix C Dimensions

Matrix \(C\) has the dimensions \(4 \times 1\), meaning it has 4 rows and 1 column.

Step 4: Validating ABC

For the product \(ABC\), multiply \(A (2 \times 4)\) by \(B (1 \times 4)\); these don't align as the inner dimensions do not match (4 ≠ 1), so \(AB\) is not defined and therefore \(ABC\) is not defined.

Step 5: Validating ACB

For \(ACB\), first calculate \(AC\), which is \(A (2 \times 4)\) by \(C (4 \times 1)\). This results in a \(2 \times 1\) matrix. Next, attempt to multiply this by \(B (1 \times 4)\), and since the dimensions (1 and 2) are not compatible, \(ACB\) is not defined.

Step 6: Validating BAC

For \(BAC\), multiply \(B (1 \times 4)\) by \(A (2 \times 4)\); the inner dimensions do not match (4 ≠ 2), so \(BA\) is not defined and \(BAC\) is not defined.

Step 7: Validating BCA

For \(BCA\), multiply \(BC\) first, where \(B (1 \times 4)\) and \(C (4 \times 1)\); this results in a \(1 \times 1\) matrix. Next, multiply the resulting \(1 \times 1\) matrix by \(A (2 \times 4)\); these dimensions don't allow for multiplication, so \(BCA\) is not defined.

Step 8: Validating CAB

For \(CAB\), multiply \(CA\) first, where \(C (4 \times 1)\) and \(A (2 \times 4)\). The inner dimensions don't allow for multiplication (1 ≠ 2), so \(CA\) is not defined and \(CAB\) is not defined.

Step 9: Validating CBA

For \(CBA\), multiply \(CB\) first, where \(C (4 \times 1)\) and \(B (1 \times 4)\); results in a \(4 \times 4\) matrix. Next, multiply this \(4 \times 4\) matrix by \(A (2 \times 4)\). These inner dimensions (4 and 2) are not compatible, so \(CBA\) is not defined.

Key Concepts

Understanding Matrix Dimensions

Matrix dimensions are like the blueprint of a matrix, indicating how many rows and columns it contains. This is crucial when performing any operation involving matrices, such as matrix multiplication. The dimensions are typically given in the form of rows by columns, written as \( m \times n \). For example, consider matrix \( A \), which has dimensions \( 2 \times 4 \). This means it has 2 rows and 4 columns. Similarly, matrix \( B \) has dimensions \( 1 \times 4 \) with 1 row and 4 columns, and matrix \( C \) has dimensions \( 4 \times 1 \) with 4 rows and 1 column.
Knowing these dimensions helps you determine if two matrices can be multiplied. It sets the groundwork for understanding more complex operations and ensures you are on the right track when computing matrix products. Remember, the number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible.

The Matrix Product

Matrix multiplication is not as straightforward as multiplying two numbers. Instead, it involves an operation called the dot product for each pair of elements between two matrices. To calculate a matrix product, matrices must be compatible in terms of their dimensions. This means the number of columns in the first matrix needs to match the number of rows in the second.
When multiplying, for example, matrix \( A (2 \times 4) \) by matrix \( C (4 \times 1) \), you multiply each element of the rows in \( A \) with the corresponding elements of the columns in \( C \), then sum these products to get the elements of the resulting matrix. The result is a matrix with dimensions derived from the rows of the first matrix and the columns of the second. In our example, \( A \times C \) results in a \( 2 \times 1 \) matrix.
It's important to carefully order the matrices when intending to multiply them because matrix multiplication is not commutative; this means \( A \times B eq B \times A \) in general. Understanding and correctly applying these rules ensures accurate calculations and valid matrix products.

The Importance of Compatibility of Matrices

Compatibility of matrices is essential for determining whether they can be multiplied, which is central in many applications across mathematics and engineering fields. When talking about compatibility, we refer to the rule that governs matrix multiplication: the number of columns of the first matrix must match the number of rows of the second matrix.
To check compatibility, take a closer look at each matrix's dimensions. For instance, with matrices \( A (2 \times 4) \) and \( C (4 \times 1) \), the inner numbers (4 in both cases) agree, so the multiplication \( A \times C \) can proceed. However, if you attempted to multiply \( A (2 \times 4) \) by \( B (1 \times 4) \) directly, the inner numbers do not match (4 and 1), and therefore, these cannot be multiplied.
Identifying compatible matrices is often a first step in solving matrix-related problems, making dimension-checking an invaluable skill in matrix algebra. Focusing on compatibility can quickly steer you towards which calculations are feasible and potentially save time by avoiding impossible operations.