Select the best answer for the question. Evaluate the determinant for the following matrix: $\left[\begin{array}{rr}3 & -5 \\ 1 & 1\end{array}\right]$ A. -4 B. 5 C. -2 D. 8
Answer (2)
Calculate the determinant using the formula: a d − b c .
Substitute the values from the matrix: ( 3 ) ( 1 ) − ( − 5 ) ( 1 ) .
Simplify the expression: 3 + 5 = 8 .
The determinant of the matrix is 8 .
Explanation
Understanding the Problem We are given a 2x2 matrix and asked to find its determinant. The matrix is [ 3 1 − 5 1 ] .
Recalling the Determinant Formula The determinant of a 2x2 matrix [ a c b d ] is calculated as a d − b c . In our case, a = 3 , b = − 5 , c = 1 , and d = 1 .
Calculating the Determinant Now, we substitute the values into the formula: ( 3 ) ( 1 ) − ( − 5 ) ( 1 ) = 3 − ( − 5 ) = 3 + 5 = 8.
Selecting the Correct Answer The determinant of the given matrix is 8. Comparing this to the given options, we see that option D is the correct answer.
Examples
Determinants are useful in various fields such as physics, engineering, and computer graphics. For example, in physics, determinants can be used to calculate the area of a parallelogram formed by two vectors. In computer graphics, they can be used to determine if a point is inside a triangle or to perform transformations such as scaling, rotation, and shearing. Understanding determinants helps in solving systems of linear equations and analyzing the properties of matrices, which are fundamental concepts in many scientific and technological applications.
The determinant of the matrix [ 3 1 − 5 1 ] is calculated as ( 3 ) ( 1 ) − ( − 5 ) ( 1 ) = 8 . Therefore, the correct answer is option D. 8.
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