Determinant From Triangular Form Lesson

In the last lesson, we learned the shortcut for finding the determinant of a 2 x 2 matrix, as well as the shortcut for finding the determinant of a 3 x 3 matrix. When we need to find the determinant of a 4 x 4 matrix or anything higher, we need to use a different technique.

Triangular Form

When we work with a square matrix, it is said to be in lower triangular form when all entries above the main diagonal are zeros. $$L=\left[ \begin{array}{ccc}a&0&0\\ d&b&0\\e&f&c\end{array}\right]$$ Similarly, an upper triangular matrix occurs when all entries below the main diagonal are zeros. $$U=\left[ \begin{array}{ccc}a&b&e\\ 0&b&f\\0&0&c\end{array}\right]$$ One useful property of a triangular matrix is that the determinant is equal to the product of the diagonal entries. Let's use this property to find the determinant of a 4 x 4 matrix.
Example #1: Find the determinant. $$A=\left[ \begin{array}{cccc}3&7&1&0\\ 1&0&2&4\\-5&-3&1&5\\1&2&0&1\end{array}\right]$$ Let's use row operations to place the matrix in upper triangular form: $$A=\left[ \begin{array}{cccc}3&7&1&0\\ 0&-\frac{7}{3}&\frac{5}{3}&4\\0&0&\frac{62}{7}&\frac{139}{7}\\0&0&0&\frac{53}{31}\end{array}\right]$$ Now our determinant is the product of the diagonal entries: $$3 \cdot -\frac{7}{3}=-7$$ $$-7 \cdot \frac{62}{7}=-62$$ $$-62 \cdot \frac{53}{31}=-2 \cdot 53=-106$$ $$|A|=-106$$

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