What is the determinant of [tex] \left[\begin{array}{lc}7 & -7 \\ 7 & 8\end{array}\right] [/tex]?
Answer (1)
The determinant of a 2x2 matrix [ a b c d ] is given by a d − b c .
Substitute the values from the given matrix: a = 7 , b = − 7 , c = 7 , and d = 8 .
Calculate the determinant: ( 7 ) ( 8 ) − ( − 7 ) ( 7 ) = 56 + 49 = 105 .
The determinant of the matrix is 105 .
Explanation
Understanding the Problem We are given a 2x2 matrix and asked to find its determinant. The determinant of a 2x2 matrix [ a b c d ] is calculated as a d − b c . In this case, the matrix is [ 7 − 7 7 8 ] , so a = 7 , b = − 7 , c = 7 , and d = 8 .
Applying the Formula To find the determinant, we substitute the values of a , b , c , and d into the formula a d − b c . This gives us ( 7 ) ( 8 ) − ( − 7 ) ( 7 ) .
Performing the Multiplication Now, we perform the multiplication: ( 7 ) ( 8 ) = 56 and ( − 7 ) ( 7 ) = − 49 . So the expression becomes 56 − ( − 49 ) .
Simplifying the Expression Next, we simplify the expression: 56 − ( − 49 ) = 56 + 49 = 105 . Therefore, the determinant of the matrix is 105.
Final Answer The determinant of the matrix [ 7 − 7 7 8 ] is 105 .
Examples
Determinants are useful in various fields such as physics, engineering, and computer graphics. For example, in physics, the determinant of a matrix can be used to calculate the area or volume of a parallelogram or parallelepiped formed by vectors. In computer graphics, determinants are used to determine if a point is inside a triangle or to calculate the area of a polygon. Understanding how to calculate determinants is essential for solving problems in these areas.
