91Ó°ÊÓ

Understanding how to multiply matrices is fundamental when working with determinants. In our exercise, although we are dealing with a 2x2 matrix determinant calculation, knowing how matrix multiplication operates can clarify why certain operations are performed.
Matrix multiplication for two 2x2 matrices involves:

• Taking the dot product of rows of the first matrix with columns of the second.
• Each element in the resulting matrix comes from these dot products.

For example, if you have \[\begin{bmatrix} e & f \ g & h \end{bmatrix} \text{ and } \begin{bmatrix} a & b \ c & d \end{bmatrix}\]the result of the multiplication will be:\[\begin{bmatrix} ea+fb & eb+fd \ gc+hd & gd+hc \end{bmatrix}\]Though we don't multiply matrices to find determinants, understanding these series of multiplication operations is crucial to grasp matrix operations as a whole.

Algebra comes into play significantly in the calculation of a 2x2 matrix's determinant. The formula for the determinant, \( ad - bc \), is a straightforward algebraic expression, but it packs a lot of power.
Let's break down this expression step by step:

• First, you multiply the diagonal elements: \( a \times d \).
• Then, multiply the off-diagonal elements: \( b \times c \).
• The determinant itself is the difference: \( ad - bc \).

Understanding this process does not only require mechanical calculation but also a comprehension of the meaning behind it. This expression evaluates how the elements of one diagonal offset the elements of the other.

Following a methodical step-by-step approach is the key to solving problems accurately, especially in mathematics like determinant calculation.
The given solution breaks down the process nicely:

• First, interpret the matrix elements as \( a = 0.75 \), \( b = -1.32 \), \( c = 0.15 \), and \( d = 1.18 \).
• Apply the determinant formula: here we substitute these elements into \( ad - bc \).
• Calculate the multiplications separately: \( 0.75 \times 1.18 = 0.885 \) and \( -1.32 \times 0.15 = -0.198 \).
• Finally, find the determinant by subtracting: \( 0.885 - (-0.198) = 1.083 \).

This stepwise breakdown helps ensure clarity and reduces the chance for errors, reinforcing the importance of organized problem-solving methods.