how to find a determinant of a 3x3 matrix
How to find a determinant of a 3x3 matrix
Answer:
To find the determinant of a (3 \times 3) matrix, you can use the cofactor expansion method. This method involves using the elements of the first row and the determinants of the (2 \times 2) submatrices. Let’s break down the process step-by-step.
Step 1: Write Down the Matrix
Consider a general (3 \times 3) matrix (A):
A = \begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
Step 2: Select the Elements of the First Row
For matrix (A), the elements of the first row are (a), (b), and (c).
Step 3: Find the Determinants of the (2 \times 2) Submatrices
For each element of the first row, you get a (2 \times 2) submatrix by removing the row and column of that element. Compute the determinants of these submatrices as follows:
-
For element (a):
Submatrix:
\begin{pmatrix} e & f \\ h & i \end{pmatrix}Determinant:
\det\begin{pmatrix} e & f \\ h & i \end{pmatrix} = ei - fh -
For element (b):
Submatrix:
\begin{pmatrix} d & f \\ g & i \end{pmatrix}Determinant:
\det\begin{pmatrix} d & f \\ g & i \end{pmatrix} = di - fg -
For element (c):
Submatrix:
\begin{pmatrix} d & e \\ g & h \end{pmatrix}Determinant:
\det\begin{pmatrix} d & e \\ g & h \end{pmatrix} = dh - eg
Step 4: Apply the Cofactor Expansion Formula
The formula for the determinant of a (3 \times 3) matrix using cofactor expansion along the first row is:
\det(A) = a \cdot \det\begin{pmatrix}
e & f \\
h & i
\end{pmatrix} - b \cdot \det\begin{pmatrix}
d & f \\
g & i
\end{pmatrix} + c \cdot \det\begin{pmatrix}
d & e \\
g & h
\end{pmatrix}
Substituting the determinants of the (2 \times 2) submatrices, we get:
\det(A) = a \cdot (ei - fh) - b \cdot (di - fg) + c \cdot (dh - eg)
Step 5: Simplify the Expression
Expand and simplify the expression to get the determinant:
\det(A) = aei + afh - bhi - bfg + cdh - ceg
Combine like terms to obtain the final result:
\det(A) = aei + afh - bhi - bfg + cdh - ceg
This expression gives the determinant of the (3 \times 3) matrix (A).
Final Answer:
The determinant of the matrix
A = \begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
is:
\boxed{\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)}
By following these steps, you can determine the determinant of any 3 \times 3 matrix.