Question

If \(A_{2 \times 3}, B_{4 \times 3}\) and \(C_{2 x 4}\) are three matrices, then which of the following is/are defined? (1) \(\mathrm{AC}^{\mathrm{T}} \mathrm{B}\) (2) \(\quad \mathrm{B}^{\mathrm{T}} \mathrm{C}^{\mathrm{T}} \mathrm{A}\) (3) \(\mathrm{AB}^{\mathrm{T}} \mathrm{C}\) (4) All the above

Short answer

Answer: The only defined multiplication is \(\mathrm{B}^{\mathrm{T}} \mathrm{C}^{\mathrm{T}} \mathrm{A}\).

Step-by-step solution

(Check \(\mathrm{AC}^{\mathrm{T}} \mathrm{B}\))

To check if \(\mathrm{AC}^{\mathrm{T}} \mathrm{B}\) is defined, first verify the matrix multiplication for each step inside the parentheses: * \(\mathrm{AC}^{\mathrm{T}}\): The matrix A has dimensions \(2\times3\) and matrix C has dimensions \(2\times4\). The transpose of matrix C, denoted by \(\mathrm{C}^{\mathrm{T}}\), would have dimensions \(4\times2\). Since the number of columns of A is not equal to the number of rows of \(\mathrm{C}^{\mathrm{T}}\), \(\mathrm{AC}^{\mathrm{T}}\) is undefined. Since the first step is undefined, the whole expression \(\mathrm{AC}^{\mathrm{T}} \mathrm{B}\) is undefined.

(Check \(\mathrm{B}^{\mathrm{T}} \mathrm{C}^{\mathrm{T}} \mathrm{A}\))

To check if \(\mathrm{B}^{\mathrm{T}} \mathrm{C}^{\mathrm{T}} \mathrm{A}\) is defined, verify matrix multiplications for each step inside the parentheses: * \(\mathrm{B}^{\mathrm{T}} \mathrm{C}^{\mathrm{T}}\): The transpose of matrix B, denoted by \(\mathrm{B}^{\mathrm{T}}\), would have dimensions \(3 \times 4\). As discussed earlier, \(\mathrm{C}^{\mathrm{T}}\) has dimensions \(4\times2\). Since the number of columns of \(\mathrm{B}^{\mathrm{T}}\) is equal to the number of rows of \(\mathrm{C}^{\mathrm{T}}\), we can multiply \(\mathrm{B}^{\mathrm{T}}\) and \(\mathrm{C}^{\mathrm{T}}\), resulting in a matrix with dimensions \(3\times2\). * Now, we need to check if the product of the previous result, with dimensions \(3\times2\), can be multiplied by matrix A of dimensions \(2\times3\). Since the number of columns in the previously obtained matrix is equal to the number of rows in matrix A, the multiplication is defined. So, \(\mathrm{B}^{\mathrm{T}} \mathrm{C}^{\mathrm{T}} \mathrm{A}\) is defined.

(Check \(\mathrm{AB}^{\mathrm{T}} \mathrm{C}\))

To check if \(\mathrm{AB}^{\mathrm{T}} \mathrm{C}\) is defined, verify matrix multiplications for each step inside the parentheses: * \(\mathrm{AB}^{\mathrm{T}}\): However, matrix A has dimensions \(2\times3\), and matrix B has dimensions \(4\times3\). Since the number of columns of A is not equal to the number of rows of B, \(\mathrm{AB}\) is undefined. Since the first step is undefined, the whole expression \(\mathrm{AB}^{\mathrm{T}} \mathrm{C}\) is undefined. Based on the analysis, the only defined multiplication is the second one, \(\mathrm{B}^{\mathrm{T}} \mathrm{C}^{\mathrm{T}} \mathrm{A}\).

Key concepts

Matrix Transpose

Matrix transpose is a fundamental operation that involves flipping a matrix over its diagonal. This means changing the row and column indices, effectively swapping the rows and columns.
Transposing a matrix can be particularly useful for addressing compatibility in matrix multiplication, as it allows the dimensions to align correctly if needed.
Considering our exercise, we saw the transpose applied, for example, to matrix C, where \[\mathrm{C}^{\mathrm{T}}\] results in switching its dimensions from \(2 \times 4\) to \(4 \times 2\).

• Transposing is denoted by the symbol \(^T\).
• The operation doesn't change the elements themselves, only their position within the matrix.
• The transpose of a transpose brings you back to the original: \( (A^T)^T = A\).

Overall, understanding matrix transpose helps in visualizing how matrices can interact through multiplication and assists in problem-solving when complex operations are involved.

Matrix Dimensions

Matrix dimensions are key to understanding how matrices can be used in various mathematical operations.
Dimensions are presented in the form \(m \times n\), where \(m\) represents the number of rows and \(n\) represents the number of columns.
When multiplying matrices, it’s crucial that the number of columns in the first matrix matches the number of rows in the second. This alignment is what allows multiplication to occur.

• The resultant matrix from a multiplication has dimensions that combine the rows of the first matrix and the columns of the second, \( m \times n \).
• If dimensions don't match up like this, multiplication cannot be performed, and the expression is undefined.

In the exercise provided, evaluating the defined or undefined nature of each multiplication was largely dependent on carefully understanding and checking these dimension rules.

Matrix Products

Matrix products involve performing calculations that multiply two matrices together. This results in a new matrix with dimensions determined by the order and sizes of the original matrices.
The fundamental rule for defining a valid matrix product is that the number of columns in the first matrix must equal the number of rows in the second matrix. If this criterion is met, each element of the resulting matrix is computed as the sum of products of elements from the rows of the first matrix and columns of the second.

• Not all matrices can be multiplied; only those satisfying the row-column condition mentioned above.
• Matrix multiplication is generally not commutative, meaning \(AB eq BA\) in most cases.

In our exercise, mastering matrix products allowed clear identification of defined expressions, as seen with \(\mathrm{B}^{\mathrm{T}} \mathrm{C}^{\mathrm{T}} \mathrm{A}\). This methodical checking is essential in ensuring correct solutions when dealing with more complex expressions.