1.2 Let
[tex]A=\left[\begin{array}{ccc}
-1 & x & -1 \\
x & -3 & 0 \\
-3 & 5 & -1
\end{array}\right][/tex]
(a) Find [tex]|A|[/tex], the determinant of [tex]A[/tex], by expanding by the second row.
(b) By using the answer obtained in (a) above, solve for [tex]x[/tex] if [tex]|A|=0[/tex].
1.3 Let
[tex]B=\left[\begin{array}{ccc}
1 & 2 & -1 \\
0 & 3 & 4 \\
1 & 7 & 2
\end{array}\right][/tex]
Determine whether [tex]B^{-1}[/tex], the inverse of [tex]B[/tex], exists. Justify your answer. (Do not determine [tex]B^{-1}[/tex]).
Answer (2)
The determinant of matrix A is ∣ A ∣ = x 2 − 5 x + 6 , and the solutions for x when ∣ A ∣ = 0 are x = 2 and x = 3 . The inverse of matrix B exists because its determinant is − 11 , which is not equal to zero.
;
Find the determinant of matrix A by expanding along the second row: ∣ A ∣ = − x 2 + 5 x + 6 .
Solve the quadratic equation − x 2 + 5 x + 6 = 0 to find x = 6 and x = − 1 .
Calculate the determinant of matrix B: ∣ B ∣ = − 11 .
Since ∣ B ∣ = 0 , the inverse of matrix B exists: B − 1 exists. B − 1 exists
Explanation
Problem Analysis We are given a matrix A and asked to find its determinant by expanding along the second row. Then, we need to solve for x when the determinant is equal to 0. Finally, we are given a matrix B and asked to determine if its inverse exists without calculating the inverse.
Determinant Calculation To find the determinant of A by expanding along the second row, we use the formula: ∣ A ∣ = − x \tC 21 + ( − 3 ) \tC 22 − 0 \tC 23 where C ij is the cofactor of the element in the i -th row and j -th column.
Calculating Cofactors Now, we calculate the cofactors C 21 and C 22 .
C 21 = ( − 1 ) 2 + 1 M 21 , where M 21 is the minor of the element in the second row and first column. M 21 = x − 1 5 − 1 = ( x ) ( − 1 ) − ( − 1 ) ( 5 ) = − x + 5 So, C 21 = ( − 1 ) 3 ( − x + 5 ) = x − 5 C 22 = ( − 1 ) 2 + 2 M 22 , where M 22 is the minor of the element in the second row and second column. M 22 = − 1 − 1 − 3 − 1 = ( − 1 ) ( − 1 ) − ( − 1 ) ( − 3 ) = 1 − 3 = − 2 So, C 22 = ( − 1 ) 4 ( − 2 ) = − 2
Finding the Determinant Substitute the cofactors into the determinant equation: ∣ A ∣ = − x ( x − 5 ) + ( − 3 ) ( − 2 ) = − x 2 + 5 x + 6
Solving for x Now, we solve for x when ∣ A ∣ = 0 :
− x 2 + 5 x + 6 = 0 x 2 − 5 x − 6 = 0 ( x − 6 ) ( x + 1 ) = 0 So, x = 6 or x = − 1
Checking for Inverse of B To determine if B − 1 exists, we need to calculate the determinant of B .
B = [ 1 2 − 1 0 3 4 1 7 2 ] ∣ B ∣ = 1 3 4 7 2 − 2 0 4 1 2 + ( − 1 ) 0 3 1 7 ∣ B ∣ = 1 ( 3 ( 2 ) − 4 ( 7 )) − 2 ( 0 ( 2 ) − 4 ( 1 )) − 1 ( 0 ( 7 ) − 3 ( 1 )) ∣ B ∣ = 1 ( 6 − 28 ) − 2 ( 0 − 4 ) − 1 ( 0 − 3 ) ∣ B ∣ = 1 ( − 22 ) − 2 ( − 4 ) − 1 ( − 3 ) ∣ B ∣ = − 22 + 8 + 3 = − 11 Since ∣ B ∣ = − 11 = 0 , the inverse of B exists.
Final Answer The determinant of matrix A is ∣ A ∣ = − x 2 + 5 x + 6 . The solutions for x when ∣ A ∣ = 0 are x = 6 and x = − 1 . The inverse of matrix B exists because its determinant is − 11 , which is not equal to 0.
Examples
Understanding determinants and matrix inverses is crucial in various fields like computer graphics, where they're used for transformations such as scaling, rotation, and translation of objects in 3D space. For instance, when designing a video game, matrix operations ensure that objects move and interact realistically within the game environment. Also, solving for variables when a determinant is zero helps identify critical conditions in systems, such as finding the points where a system of linear equations has no unique solution, which is vital in engineering and physics.
