91Ó°ÊÓ

Matrix multiplication is a fundamental operation in linear algebra. Unlike element-wise multiplication, matrix multiplication involves a row-by-column product. For two matrices, say \( \mathbf{A} \) and \( \mathbf{B} \), to be multiplicable, the number of columns in \( \mathbf{A} \) must be equal to the number of rows in \( \mathbf{B} \).
The resulting matrix, known as the product matrix, has dimensions determined by the rows of the first and columns of the second matrix involved in the operation.

The process can be thought of as taking a row from the first matrix and a column from the second matrix, then multiplying the corresponding elements and adding them together to form a single entry in the product matrix.

In the given exercise, matrix multiplication occurs when we calculate \( \mathbf{A}^{2} \) by multiplying matrix \( \mathbf{A} \) with itself. The operation is conducted by following the row-by-column rule for each element of the resulting matrix \( \mathbf{A}^{2} \).

The identity matrix, often symbolized as \( \mathbf{I} \), plays a crucial role in matrix operations, serving as the multiplicative identity in the matrix world. This means any matrix \( \mathbf{A} \) multiplied by an identity matrix \( \mathbf{I} \) results in the original matrix \( \mathbf{A} \).
An identity matrix is always square with a size of \( n \times n \) and it has ones on the diagonal and zeros everywhere else. The pattern looks like this for a 3x3 identity matrix: \[ \left(\begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right) \]

In the exercise, \( 2 \mathbf{I} \) refers to each element of the identity matrix being multiplied by 2. This operation does not change the position of the ones but rather scales them by the factor provided.

Matrix addition is another fundamental operation in which two matrices of the same dimensions are added together element-wise. The rule is simple: just add the numbers that are in the same position in each matrix.

For example, if you have two matrices \( \mathbf{A} \) and \( \mathbf{B} \) of the same size, their sum \( \mathbf{A} + \mathbf{B} \) would result in a matrix where each element \( c_{ij} \) is the sum of elements \( a_{ij} \) and \( b_{ij} \) from the original matrices.

In our exercise, the final step requires us to add the matrices \( \mathbf{A^2} \), \( 3 \mathbf{A} \) and \( 2 \mathbf{I} \) together, following this element-wise addition rule. The resultant matrix combines the values from each of these matrices into one final matrix.