Given the matrix \( \begin{pmatrix} 3 & 10 & 6 \ -6 & 0 & 15 \ -2 & 3 & 12 \ \end{pmatrix} \), what operations can be performed on it?
Understand the Problem
The question appears to be asking for analysis or calculations involving the given matrix. This could include finding the determinant, inverse, or performing matrix operations. We will focus on determining the type of mathematical operation required.
Answer
The determinant of the matrix is \( \text{det}(A) = 177 \).
Answer for screen readers
The determinant of the matrix is ( \text{det}(A) = 177 ).
Steps to Solve
-
Identify the Operation
Firstly, identify the operation we need to perform on the given matrix. In this case, let's find the determinant. -
Setup the Determinant Formula
For a (3 \times 3) matrix, the determinant can be calculated using the formula:
$$ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) $$
Where the matrix elements are arranged as follows:
$$
\begin{pmatrix}
a & b & c \
d & e & f \
g & h & i
\end{pmatrix}
$$
So for the given matrix:
$$
\begin{pmatrix}
3 & 10 & 6 \
-6 & 0 & 15 \
-2 & 3 & 12
\end{pmatrix}
$$
we have (a = 3), (b = 10), (c = 6), (d = -6), (e = 0), (f = 15), (g = -2), (h = 3), and (i = 12).
-
Calculate Each Part of the Determinant
Now, we evaluate the terms in the formula:
-
First term:
$$ 3(0 \cdot 12 - 15 \cdot 3) = 3(0 - 45) = 3 \cdot (-45) = -135 $$ -
Second term:
$$ -10(-6 \cdot 12 - 15 \cdot -2) = -10(-72 + 30) = -10 \cdot -42 = 420 $$ -
Third term:
$$ 6(-6 \cdot 3 - 0 \cdot -2) = 6(-18 - 0) = 6 \cdot (-18) = -108 $$
-
Combine the Results to Find the Determinant
Adding all these results gives:
$$
\text{det}(A) = -135 + 420 - 108
$$
Calculating this results in:
$$ \text{det}(A) = 177 $$
The determinant of the matrix is ( \text{det}(A) = 177 ).
More Information
The determinant is a scalar value that provides important properties about the matrix, including whether it is invertible. A non-zero determinant indicates that the matrix has an inverse.
Tips
- Confusing the signs of the terms while calculating the determinant. It's important to pay attention to the order of operations and signs indicated in the formula.
- Failing to multiply the correct elements when setting up the determinant calculation. Always ensure that the correct matrix elements are used per the formula.
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