Compute the determinant of matrix A = [16, 19, 13; 14, 12, 19; 15, 1, 6].

To compute the determinant of matrix \( A = \begin{bmatrix} 16 & 19 & 13 \\ 14 & 12 & 19 \\ 15 & 1 & 6 \end{bmatrix} \), we will apply the formula for the determinant of a 3x3 matrix:

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

where the elements of the matrix are:

\[
A = \begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix}
\]

For matrix \( A \), we have:
- \( a = 16 \), \( b = 19 \), \( c = 13 \)
- \( d = 14 \), \( e = 12 \), \( f = 19 \)
- \( g = 15 \), \( h = 1 \), \( i = 6 \)

Now, calculating step-by-step:

\[
\text{det}(A) = 16(12 \times 6 - 19 \times 1) - 19(14 \times 6 - 19 \times 15) + 13(14 \times 1 - 12 \times 15)
\]

Let me compute this.The determinant of matrix \( A = \begin{bmatrix} 16 & 19 & 13 \\ 14 & 12 & 19 \\ 15 & 1 & 6 \end{bmatrix} \) is \( \text{det}(A) = 2509 \).

Would you like a detailed breakdown of the steps? Here are 5 related questions you might want to explore:

1. How does the determinant help in solving systems of linear equations?
2. What does the sign of a determinant indicate about a matrix?
3. How can the determinant of a matrix be used to find its inverse?
4. What are some properties of determinants that simplify their calculation?
5. How would you compute the determinant of a larger matrix, such as 4x4?

**Tip:** A matrix with a determinant of zero is called singular, meaning it doesn't have an inverse.

Learn how to compute the determinant of matrix A = [16, 19, 13; 14, 12, 19; 15, 1, 6] using the determinant formula for 3x3 matrices. This solution covers key concepts from linear algebra, including matrix properties and determinant calculation. Suitable for students in Grades 10-12.

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