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A determinant provides a scalar value that summarizes certain properties of a matrix. For instance, it helps us determine if a matrix is invertible. Calculating the determinant of a 3x3 matrix can follow a specific formula:\[ \text{det}(\boldsymbol{A}) = a(ei-fh) - b(di-fg) + c(dh-eg) \]In this formula, each letter represents elements of the matrix placed in respective positions. **To start**, let's apply it to matrix \( \boldsymbol{A} \). By substituting in the values:\[ \text{det}(\boldsymbol{A}) = 1(4 \cdot 1 - 0 \cdot -2) - 0(6 \cdot 1 - 0 \cdot 6) + 2(6 \cdot -2 - 4 \cdot 6) \]This reduces to \( \text{det}(\boldsymbol{A}) = 4 + 0 - 72 = -68 \). A non-zero determinant, such as \(-68\), signifies that the matrix \( \boldsymbol{A} \) is invertible.

The adjugate matrix, also known as the adjunct or adjoint, is essential in constructing the inverse of a matrix. It is obtained by taking the transpose of the cofactor matrix. Here's a simple guide:

• **Cofactor matrix:** Each element is the determinant of a smaller matrix, called a minor, obtained by deleting the current row and column.
• **Transpose:** Flip the matrix over its diagonal.

For instance, for matrix \( \boldsymbol{A} \), the adjugate is:\[ \text{adj}(\boldsymbol{A}) = \begin{bmatrix} 4 & 0 & 12 \ -6 & -11 & -6 \ -8 & 12 & 4 \end{bmatrix} \]Generating the adjugate correctly ensures accurate determination of the matrix inverse.

Matrix multiplication is a process used to combine two matrices into a single matrix. Each element is computed by multiplying entries of the row of the first matrix by corresponding entries of the column of the second matrix. For matrices \( \boldsymbol{A} \) and \( \boldsymbol{B} \), we multiply to get \( \boldsymbol{A}\boldsymbol{B} \):\[ \boldsymbol{A}\boldsymbol{B} = \begin{bmatrix} 1 & 0 & 2 \ 6 & 4 & 0 \ 6 & -2 & 1 \end{bmatrix} \begin{bmatrix} 5 & 2 & 4 \ 3 & -1 & 2 \ 1 & 4 & -3 \end{bmatrix} = \begin{bmatrix} 7 & 10 & -2 \ 42 & 4 & 28 \ 29 & 0 & 19 \end{bmatrix} \]This operation is not commutative; swapping matrices\( \boldsymbol{A} \) and \( \boldsymbol{B} \) will lead to a different result.

One intriguing property of matrices is how inverses behave under multiplication. Specifically, for any two invertible matrices \( \boldsymbol{A} \) and \( \boldsymbol{B} \), their product's inverse \((\boldsymbol{A}\boldsymbol{B})^{-1}\) equals the product of their inverses in reverse order: \(\boldsymbol{B}^{-1}\boldsymbol{A}^{-1}\). To verify:

• First compute \(\boldsymbol{A}\boldsymbol{B}\).
• Find the inverse \((\boldsymbol{A}\boldsymbol{B})^{-1}\).
• Separately calculate \(\boldsymbol{B}^{-1}\) and \(\boldsymbol{A}^{-1}\), then multiply them.
• Compare \((\boldsymbol{A}\boldsymbol{B})^{-1}\) with \(\boldsymbol{B}^{-1}\boldsymbol{A}^{-1}\).

In our exercise, performing these steps confirmed the property. Thus, understanding this principle is essential for efficiently handling complex matrix calculations.