What is the determinant of the matrix $$\begin{pmatrix} 0 & 7 & 3 \\ 1 & 0 & 7 \\ 3 & 0 & 0 \end{pmatrix}$$ ? Simplify your answer. (determinant, matrix, linear algebra, cofactor expansion, square matrix)

Step 1: Choose a row or column for cofactor expansion

To find the determinant of a $3 \times 3$ matrix, we can use the cofactor expansion method. This involves choosing a row or a column and then calculating the sum of the products of each element with its corresponding cofactor. It's generally easiest to choose a row or column that contains the most zeros, as this simplifies calculations. In this matrix, the third row and the second column both have two zeros.

Step 2: Expand along the third row

Let's expand along the third row. The elements are $a_{31}=3$, $a_{32}=0$, and $a_{33}=0$. The cofactor for an element $a_{ij}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by removing the $i$ -th row and $j$ -th column. Since the second and third elements of the third row are zero, their contributions to the determinant will be zero, simplifying the calculation significantly.

Step 3: Calculate the determinant of the $2 \times 2$ submatrix

Now we need to calculate the determinant of the $2 \times 2$ submatrix: $\begin{vmatrix} 7 & 3 \\ 0 & 7 \end{vmatrix}$. The determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by $ad - bc$.

Step 4: Substitute and simplify

Substitute the determinant of the $2 \times 2$ submatrix back into the expansion formula. The terms involving zeros will vanish, leaving only the product of $3$, $1$ (from $(-1)^{3+1}$), and the determinant of the submatrix.

Step 5: Final calculation

Perform the final multiplication to get the determinant of the matrix.

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