91Ó°ÊÓ

An invertible matrix is a square matrix that has an inverse, meaning there exists another matrix which, when multiplied with the original matrix, yields the identity matrix. For a square matrix \( A \), if there exists a matrix \( A^{-1} \) such that:

• \( A A^{-1} = I \)
• \( A^{-1} A = I \)

where \( I \) is the identity matrix, then \( A \) is invertible or non-singular. Invertibility is crucial because it allows us to "undo" matrix operations, much like how dividing by a number "undoes" multiplication. If a matrix is invertible, you can solve equations involving the matrix, like the one in the exercise where we solved \( B A = C A \) by multiplying both sides by \( A^{-1} \). This step was key in showing that \( B = C \). Without an inverse, we're stuck, as the matrix multiplication cannot be reversed.

The identity matrix is a special kind of matrix that acts like the number 1 in matrix multiplication. It's a square matrix with ones on the diagonal and zeros elsewhere, typically denoted as \( I \). For any matrix \( A \) of the same size, multiplying by the identity matrix does not change \( A \):

• \( A I = I A = A \)

Using identity matrices is fundamental when working with inverses, because it confirms that an inverse returns us back to where we started. Think of identity matrices as the 'do nothing' operation in matrix multiplication. In our exercise, we utilized the property \( A A^{-1} = I \) to simplify the equation \( B (A A^{-1}) = C (A A^{-1}) \) to \( B I = C I \), finally leading us to \( B = C \). This simplicity comes from the identity matrix's neutral role in multiplication.

Matrix multiplication is a method of combining two matrices to produce a new matrix. Unlike simple arithmetic, matrix multiplication isn't commutative, meaning \( AB \) does not necessarily equal \( BA \). In multiplication:

• Each element of the resulting matrix is computed as the dot product of a row of the first matrix and a column of the second matrix
• The number of columns in the first matrix must equal the number of rows in the second matrix

In our exercise, matrix multiplication allowed us to manipulate the equation \( B A = C A \). By using properties like associativity, where \( (AB)C = A(BC) \), we managed to simplify the expressions and solve the equation efficiently. Mastery of matrix multiplication is essential to understanding many linear algebra procedures.

A counterexample in mathematics is an example that disproves a statement or proposition. It demonstrates the fallacy of a statement by showing a single instance where the statement is not true, sometimes under modified conditions. In part (b) of our exercise, we needed to show a case where the conclusion from part (a) doesn't hold if \( A \) is not invertible.We used \( A = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} \), a zero matrix, which isn't invertible, as our counterexample. Given \( B = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \) and \( C = \begin{bmatrix} 2 & 0 \ 0 & 2 \end{bmatrix} \), both \( B A \) and \( C A \) result in a zero matrix, confirming \( B A = C A \) even though \( B eq C \). This illustrates that without the condition of invertibility, our previous conclusions do not necessarily apply.