Exercise 29 Page 195

To evaluate the determinant of a 3* 3 matrix, we use the diagonal rule.

1. Rewrite the first two columns to the right of the determinant.
2. Draw diagonals, beginning with the upper-left number. Multiply the numbers in each diagonal.
3. Repeat the second step, this time beginning with the bottom-left number.
4. Find the sum of the products of the numbers in each set of diagonals, and subtract the second sum from the first.

Let's do it!

Step 1

We will write the given determinant and copy the first two columns on the right-hand side.

Step $2$

Now, we will draw diagonals beginning with the upper-left number.

Let's multiply the numbers in each diagonal. \begin{aligned} \col{3}\t(\textcolor{#FF8C00}{\N 4})\t 5 &= \colIII{\N 60} \\ \textcolor{#FF8C00}{5}\t 6 \t(\col{\N 6}) &= \colIII{\N 180} \\ \N 2\t(\col{\N 1})\t(\textcolor{#FF8C00}{\N 2}) &= \colIII{\N 4} \end{aligned}

Step $3$

We will repeat the previous step, but draw diagonals beginning with the bottom-left number.

As we did before, let's multiply the numbers in each diagonal. \begin{aligned} \col{\N 6}\t(\textcolor{#FF8C00}{\N 4})\t(\N 2) &= \colII{\N 48} \\ \textcolor{#FF8C00}{\N 2}\t 6 \t \col{3} &= \colII{\N 36} \\ 5\t(\col{\N 1})\t \textcolor{#FF8C00}{5} &= \colII{\N 25} \end{aligned}

Step $4$

Finally, we will find the sum of the products in each set of diagonals. Then we will subtract the second sum from the first sum.