The question asks us to find the value of a given 3x3 determinant. Calculating the determinant of a matrix is a fundamental operation in linear algebra. For a 3x3 matrix, we can use the cofactor expansion method along any row or column. Let's use the first row for expansion.
The given determinant is:
\(
\begin{vmatrix}
\cos \rm C & \tan \rm A & 0\\
\sin \rm B & 0 & -\tan \rm A\\
0 & \sin \rm B & \cos \rm C
\end{vmatrix}
\)
Calculating the Determinant Value Step-by-Step
To calculate the determinant of a 3x3 matrix \(\begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}\), we use the formula:
The first term: \(\cos \rm C\) * (0 - (\(-\tan \rm A \sin \rm B\))) = \(\cos \rm C\) * (\(\tan \rm A \sin \rm B\)) = \(\tan \rm A \sin \rm B \cos \rm C\)
The second term: \(-\tan \rm A\) * (\(\sin \rm B \cos \rm C\) - 0) = \(-\tan \rm A\) * (\(\sin \rm B \cos \rm C\)) = \(-\tan \rm A \sin \rm B \cos \rm C\)
The third term: 0 * ((\(\sin \rm B\)) * (\(\sin \rm B\)) - 0) = 0
Adding these terms together to get the total determinant value:
Determinant = \(\tan \rm A \sin \rm B \cos \rm C\) - \(\tan \rm A \sin \rm B \cos \rm C\) + 0
Determinant = 0
Thus, the value of the determinant is 0.
Summary of Determinant Calculation
By expanding the determinant along the first row, we calculated the value by summing the products of each element and its corresponding cofactor. The calculation resulted in the expression \(\tan \rm A \sin \rm B \cos \rm C - \tan \rm A \sin \rm B \cos \rm C\), which simplifies to 0.
Element
Cofactor Sign
Minor Determinant
Product (Element * Cofactor)
\(\cos \rm C\)
+
\(\begin{vmatrix} 0 & -\tan \rm A \\ \sin \rm B & \cos \rm C \end{vmatrix} = 0 \cdot \cos \rm C - (-\tan \rm A) \cdot \sin \rm B = \tan \rm A \sin \rm B\)
\(\cos \rm C (\tan \rm A \sin \rm B) = \tan \rm A \sin \rm B \cos \rm C\)
\(\tan \rm A\)
-
\(\begin{vmatrix} \sin \rm B & -\tan \rm A \\ 0 & \cos \rm C \end{vmatrix} = \sin \rm B \cdot \cos \rm C - (-\tan \rm A) \cdot 0 = \sin \rm B \cos \rm C\)
\(-\tan \rm A (\sin \rm B \cos \rm C) = -\tan \rm A \sin \rm B \cos \rm C\)
0
+
\(\begin{vmatrix} \sin \rm B & 0 \\ 0 & \sin \rm B \end{vmatrix} = \sin \rm B \cdot \sin \rm B - 0 \cdot 0 = \sin^2 \rm B\)
\(0 (\sin^2 \rm B) = 0\)
Sum of products = \(\tan \rm A \sin \rm B \cos \rm C - \tan \rm A \sin \rm B \cos \rm C + 0 = 0\)
Revision Table: Determinant Calculation
Concept
Description
Formula/Method
Determinant
A scalar value that can be computed from the elements of a square matrix.
Specific formulas for 2x2, 3x3, and higher order matrices.
3x3 Determinant Calculation
Using cofactor expansion along a row or column.
\(\begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)\)
Cofactor
The signed minor of an element. Sign depends on position \((-1)^{i+j}\).
\(C_{ij} = (-1)^{i+j} M_{ij}\)
Minor
The determinant of the submatrix formed by deleting the row and column of the element.
\(M_{ij}\) for element at row i, column j.
Additional Information: Properties of Determinants
Understanding the properties of determinants can sometimes simplify their calculation or provide insights without direct computation. Some key properties include:
If a row or column consists entirely of zeros, the determinant is 0.
If two rows or two columns are identical, the determinant is 0.
If one row or column is a multiple of another row or column, the determinant is 0.
Swapping two rows or columns multiplies the determinant by -1.
Multiplying a row or column by a scalar \(k\) multiplies the determinant by \(k\).
Adding a multiple of one row (or column) to another row (or column) does not change the determinant.
The determinant of a transpose of a matrix is equal to the determinant of the original matrix (\(\det(A^T) = \det(A)\)).
The determinant of a product of matrices is the product of their determinants (\(\det(AB) = \det(A)\det(B)\)).
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