91Ó°ÊÓ

An invertible matrix is a square matrix that has an inverse. This means if you multiply the matrix by its inverse, the result will be the identity matrix. Notably, a matrix is invertible if and only if its determinant is non-zero. For example, consider a matrix \(A\). If \( \text{det}(A) eq 0 \), then \(A\) is invertible.

In this exercise, after computing the determinant of the given matrix as \(-142\), we determined that the matrix is invertible because \(-142 eq 0\). Recognizing if a matrix is invertible without computing the reverse can save a lot of time for larger matrices.

A 3x3 matrix is a square matrix with three rows and three columns. In notation, it appears as:

\[\begin{array}{rrr} a & b & c \ d & e & f \ g & h & i \end{array}\]

Each element in the matrix is denoted by a letter, where the letters themselves stand for numbers. For the given exercise, the 3x3 matrix is:

\[\left[\begin{array}{rrr} -1 & 4 & 0 \ 0 & 3 & -2 \ 5 & 8 & 7 \end{array}\right]\]

The structure of the 3x3 matrix means it has more elements to consider than smaller matrices like 2x2, making determinant calculations a bit more complex. However, following a clear methodical approach, as shown in the solution, simplifies these calculations considerably.

The determinant of a matrix is a special number that gives a lot of information about the matrix, including whether it's invertible. For a 3x3 matrix \(\left[ \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right]\), the determinant is calculated using the formula:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

In our given matrix:

\[ \left[\begin{array}{rrr} -1 & 4 & 0 \ 0 & 3 & -2 \ 5 & 8 & 7 \end{array}\right] \],

we substitute as follows:

\[ \text{det}(A) = (-1)((3 \times 7) - (-2 \times 8)) - 4((0 \times 7) - (-2 \times 5)) + 0((0 \times 8) - (3 \times 5)) \]

Simplify each portion:

\[ = -1(21 + 16) - 4(0 + 10) + 0 \]

Calculate individually:

\[ = -1 \times 37 - 4 \times 10 + 0 \]

Combine results:

\[ = -37 - 40 \]

Finally:

\[ = -77 \]

Thus, the determinant of the given matrix is \(-77\), confirming that the matrix is invertible.