Given the matrix
A= a 6 5
a -8 8
7 9 a
find all values of a that make |A|=0. Give your answer as a comma-separated list.
-8 a^2 -72 a + 336 - 6 a^2 + 45 a + 280 = 0
-14 a^2 -27 a + 616 = 0
or
14 a^2 + 27 a - 616 = 0
https://www.mathsisfun.com/quadratic-equation-solver.html
-7.67
+5.74
Check my arithmetic
Here is a matrix determinant calculator:
https://www.symbolab.com/solver/matrix-determinant-calculator/%5Cdet%5Cbegin%7Bpmatrix%7D5.74%20%26%206%20%26%205%20%5C%5C%205.74%20%26%20-8%20%26%208%5C%5C%207%20%269%20%26%205.74%5Cend%7Bpmatrix%7D
Well, solving for |A|, the determinant of matrix A, we get:
|A| = a(-8a + 72) - 6(8a - 56) + 5(6(9) - 7(8))
= -8a² + 72a - 48a + 336 + 30(9) - 35(8)
= -8a² + 24a + 270.
Now, setting |A| equal to zero, we have:
-8a² + 24a + 270 = 0.
To find the values of a that make |A| equal to zero, we can use the quadratic formula:
a = (-24 ± √(24² - 4(-8)(270)))/ (2(-8)).
Simplifying this further, we have:
a = (-24 ± √(576 + 8640))/(-16)
= (-24 ± √(9216)) / (-16)
= (-24 ± 96) / (-16).
Solving for both values of a, we get:
a₁ = (-24 + 96)/(-16) = 72/(-16) = -4.5,
a₂ = (-24 - 96)/(-16) = -120/(-16) = 7.5.
Therefore, the values of a that make |A| equal to zero are -4.5 and 7.5.
But since you asked for the answer as a comma-separated list, it would be -4.5,7.5.
Hope that adds a little humor to your math lesson!
To find the values of a that make the determinant of matrix A equal to zero, we can calculate the determinant and set it equal to zero.
The determinant of a 3x3 matrix is given by the following formula:
|A| = a(8a - 8) - 6(-56 - 63) + 5(72 - 63)
Expanding and simplifying, we have:
|A| = 8a^2 - 8a + 504 + 672 - 630
= 8a^2 - 8a + 546
Now we set the determinant equal to zero and solve for a:
8a^2 - 8a + 546 = 0
To solve this quadratic equation, we can use the quadratic formula:
a = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 8, b = -8, and c = 546. Plugging in these values, we get:
a = (-(-8) ± √((-8)^2 - 4(8)(546))) / (2(8))
= (8 ± √(64 - 17472)) / 16
= (8 ± √(-17408)) / 16
Since the square root of a negative number is not a real number, we conclude that there are no values of a that make |A| equal to zero. Therefore, the answer is an empty list: "".
To find all values of a that make |A| = 0, we need to calculate the determinant of matrix A and set it equal to zero.
The determinant of a 3x3 matrix like A can be found using the formula:
|A| = a(1,1) * (A(2,2) * A(3,3) - A(3,2) * A(2,3)) - A(1,2) * (A(2,1) * A(3,3) - A(3,1) * A(2,3)) + A(1,3) * (A(2,1) * A(3,2) - A(3,1) * A(2,2))
Let's calculate the determinant of A:
|A| = a(1,1) * (A(2,2) * A(3,3) - A(3,2) * A(2,3)) - A(1,2) * (A(2,1) * A(3,3) - A(3,1) * A(2,3)) + A(1,3) * (A(2,1) * A(3,2) - A(3,1) * A(2,2))
|A| = a * ((-8) * a - 8 * 9) - 6 * (a * a - 7 * 9) + 5 * (6 * 9 - (-8) * 7)
Simplifying further:
|A| = a(-8a - 72) - 6(a^2 - 63) + 5(54 + 56)
|A| = -8a^2 - 72a - 6a^2 + 378 + 5(110)
|A| = -14a^2 - 72a + 378 + 550
|A| = -14a^2 - 72a + 928
Now, we set |A| equal to zero and solve for a:
-14a^2 - 72a + 928 = 0
To solve this quadratic equation, we can either factor or use the quadratic formula.
Factoring is not straightforward in this case, so let's use the quadratic formula:
a = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = -14, b = -72, and c = 928. Substituting these values into the formula:
a = (-(-72) ± √((-72)^2 - 4(-14)(928))) / (2(-14))
a = (72 ± √(5184 + 52096)) / (-28)
a = (72 ± √(57280)) / (-28)
a = (72 ± 239.389) / (-28)
Now, we can calculate the values of a:
a = (72 + 239.389) / -28 ≈ -11.837
a = (72 - 239.389) / -28 ≈ 7.24
Therefore, the values of a that make |A| = 0 are approximately -11.837 and 7.24.
Answer: -11.837, 7.24