If \(A = \left( {\begin{array}{*{20}{c}} 1&2\\ 2&3\\ 3&4 \end{array}} \right)\) and \(B = \left( {\begin{array}{*{20}{c}} 1&2\\ 2&1 \end{array}} \right),\) then which one of the following is correct?
AB exists but BA does not exist
Understanding Matrix Product Existence
This question asks us to determine whether the matrix products AB and BA exist, given the matrices A and B. The existence of a matrix product depends on the dimensions of the matrices being multiplied.
Analyzing Matrix Dimensions
First, let's identify the dimensions of the given matrices:
- Matrix \(A = \left( {\begin{array}{*{20}{c}} 1&2\\ 2&3\\ 3&4 \end{array}} \right)\) has 3 rows and 2 columns. So, the dimension of matrix A is \(3 \times 2\).
- Matrix \(B = \left( {\begin{array}{*{20}{c}} 1&2\\ 2&1 \end{array}} \right)\) has 2 rows and 2 columns. So, the dimension of matrix B is \(2 \times 2\).
Condition for Matrix Multiplication
For the product of two matrices, say C and D, to exist (i.e., CD exists), the number of columns in the first matrix (C) must be equal to the number of rows in the second matrix (D).
If matrix C has dimensions \(m \times n\) and matrix D has dimensions \(p \times q\), the product CD exists only if \(n = p\). If it exists, the resulting matrix CD will have dimensions \(m \times q\).
Checking the Existence of AB
To check if the product AB exists, we look at the dimensions of A and B in that order:
- Dimension of A is \(3 \times 2\). Number of columns in A is 2.
- Dimension of B is \(2 \times 2\). Number of rows in B is 2.
Since the number of columns in A (2) is equal to the number of rows in B (2), the matrix product AB exists. The dimension of the resulting matrix AB would be \(3 \times 2\).
Checking the Existence of BA
To check if the product BA exists, we look at the dimensions of B and A in that order:
- Dimension of B is \(2 \times 2\). Number of columns in B is 2.
- Dimension of A is \(3 \times 2\). Number of rows in A is 3.
Since the number of columns in B (2) is not equal to the number of rows in A (3), the matrix product BA does not exist.
Conclusion on Matrix Products AB and BA
Based on the analysis of the dimensions, we found that:
- The product AB exists.
- The product BA does not exist.
Therefore, the correct statement is that AB exists but BA does not exist.
Comparing with the Options
Let's compare our findings with the given options:
- Both AB and BA exist - Incorrect, as BA does not exist.
- Neither AB nor BA exists - Incorrect, as AB exists.
- AB exists but BA does not exist - Correct, this matches our finding.
- AB does not exist but BA exists - Incorrect, as AB exists and BA does not exist.
The option that correctly describes the existence of the matrix products AB and BA is "AB exists but BA does not exist".
Revision Table: Matrix Multiplication Condition
| Product | First Matrix Dimension | Second Matrix Dimension | Condition for Existence | Resulting Dimension (if exists) |
|---|---|---|---|---|
| CD | \(m \times n\) | \(p \times q\) | Number of columns in C = Number of rows in D (\(n = p\)) | \(m \times q\) |
| AB | A (\(3 \times 2\)) | B (\(2 \times 2\)) | Columns of A (2) = Rows of B (2)? Yes. | \(3 \times 2\) |
| BA | B (\(2 \times 2\)) | A (\(3 \times 2\)) | Columns of B (2) = Rows of A (3)? No. | Does not exist |
Additional Information on Matrix Multiplication
Matrix multiplication is generally not commutative. This means that for two matrices A and B, even if both AB and BA exist, they are usually not equal (i.e., AB \(\neq\) BA). In this specific problem, we see an extreme case where one product (AB) exists, but the other (BA) doesn't exist at all, further highlighting the non-commutative nature of matrix multiplication.
Understanding matrix dimensions and the condition for multiplication is fundamental in linear algebra. It allows us to determine when matrix operations are valid and predict the size of the resulting matrices.
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