\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix}

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Step-by-Step Solution

Step 1: Define the matrix and determinant formula

We are given a 3x3 matrix. To find its determinant, we use the cofactor expansion method. The formula involves multiplying each element of the first row by the determinant of its corresponding 2x2 submatrix, with alternating signs.

Step 2: Substitute matrix elements into the formula

Now, we substitute the values from the given matrix into the determinant formula. For the first term, we take $a_{11}=1$ and multiply it by the determinant of the submatrix $\begin{pmatrix} 1 & 4 \\ 6 & 0 \end{pmatrix}$. We repeat this for $a_{12}=2$ and $a_{13}=3$, remembering the alternating signs.

Step 3: Calculate the determinants of the 2x2 submatrices

Next, we calculate the determinant for each of the 2x2 submatrices. The determinant of a 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by $ad - bc$.

Step 4: Perform the multiplications and additions

Now, we perform the multiplications for each term. Then, we combine these results through addition and subtraction to find the final determinant.

Step 5: Calculate the final determinant

Finally, we sum the values: $-24 + 40 - 15 = 1$. This gives us the determinant of the given matrix.