91Ó°ÊÓ

Assume that \(A\) is a reducible \(2 \times 2\) matrix. Then, \(A\) can be reduced to its canonical form. Using different cases, we can conclude that \(a_{12} a_{21} = 0\): Case 1: If \(a_{11}=0\), then we can perform an elementary column operation to swap columns 1 and 2. This column operation does not affect the product \(a_{12} a_{21}\). In this case, we can perform a row operation to make the leading coefficent of the first row equal to 1. If the leading coefficient becomes 1, then the second element in the first row should be 0 for the matrix to be be fully canonical, which forces \(a_{12}=0\) and \(a_{12} a_{21} = 0\). Case 2: If \(a_{11}\neq0\), then we can use an elementary row operation to make the leading coefficient of the first row equal to 1. In this case, we need to make the second element in the first row 0. Hence we should have the second element in the second row (\(a_{21}\)) nonzero to make the first element in the second row 0. But for the canonical form of a \(2 \times 2\) matrix, we should also have the first element in the second row equal to 0, i.e., \(a_{21} = 0\). Thus, \(a_{12} a_{21} = 0\).

Now, we need to prove the converse, i.e., if the product \(a_{12} a_{21} = 0\), then \(A\) can be reduced to the canonical form. We can prove this by contradiction. Suppose \(A\) is not reducible. This means we cannot transform \(A\) into a canonical form using elementary row and column operations. Case 1: If \(a_{12}=0\), then we need to show \(A\) can be reduced to the canonical form or \(a_{21}\neq0\). But, by assumption, \(a_{12}a_{21}=0\). Since \(a_{12}=0\), we must have \(a_{21}=0\), contradicting our assumption that \(A\) is not reducible. Case 2: If \(a_{21}=0\), then we need to show \(A\) can be reduced to the canonical form or \(a_{12}\neq0\). But, by assumption, \(a_{12}a_{21}=0\). Since \(a_{21}=0\), we must have \(a_{12}=0\), contradicting our assumption that \(A\) is not reducible. Therefore, by contradiction, \(A\) must be reducible if \(a_{12} a_{21} = 0\). Since we have proven both implications, we conclude that a \(2\times2\) matrix \(A\) is reducible if and only if \(a_{12} a_{21}=0\).