Question

Identify the inverse of the given matrix, if it exists. [[3,-4],[5,-8]]

Answer 1

The inverse of the matrix [[3, -4], [5, -8]] exists and is calculated by taking 1 divided by the determinant and multiplying it by the adjoint of the matrix. The resulting inverse is [[2, -1], [1.25, -0.75]].

Step-by-step explanation:

The student has asked to identify the inverse of a 2x2 matrix, which is given by the formula:

If the matrix is A = [[a, b], [c, d]], then the inverse of A, denoted as A-1, is 1/(ad - bc) * [[d, -b], [-c, a]], provided that ad - bc is not equal to zero.

Applying this formula to the given matrix A = [[3, -4], [5, -8]], first we calculate the determinant (ad - bc), which is (3)(-8) - (-4)(5) = -24 - (-20) = -24 + 20 = -4.

Since the determinant is not zero, the inverse exists, and it is:

A-1 = 1/(-4) * [[-8, 4], [-5, 3]] = [[2, -1], [1.25, -0.75]]

Answer 2

The inverse of the matrix [[3,-4],[5,-8]] exists and is computed by first calculating its determinant (which is not zero) and then applying the formula for the inverse of a 2x2 matrix. The result is the matrix [[2, -1], [1.25, -0.75]].

Step-by-step explanation:

To find the inverse of a 2x2 matrix, you can use the formula:

If the given matrix A is:

A = [[a, b], [c, d]]

The inverse of matrix A, denoted A-1, is given by:

A-1 = (1/det(A)) * [[d, -b], [-c, a]], where det(A) is the determinant of matrix A.

First, we calculate the determinant:

det(A) = (3)(-8) - (-4)(5) = -24 - (-20) = -24 + 20 = -4

Since the determinant is not zero, the inverse exists. Now we can calculate the inverse using the formula:

A-1 = (1 / -4) * [[-8, 4], [-5, 3]]

A-1 = [[2, -1], [1.25, -0.75]]

So, the inverse of the matrix [[3,-4],[5,-8]] is [[2, -1], [1.25, -0.75]]