Question

A matrix a is given. determine if the system ax = b (where x and b have the appropriate number of components) has a solution for all choices of

b. a = 5 −4 4 2

Answer 1

The determinant of the given matrix a is 26, which is not zero; thus, the matrix is invertible, indicating that the system ax = b has a solution for every vector b.

Step-by-step explanation:

To determine if the system ax = b has a solution for all choices of b, we need to look at the properties of the matrix a. The matrix given is:

\[ \begin{bmatrix} 5 & -4 \\ 4 & 2 \end{bmatrix} \]

We want to determine if this matrix is invertible, which would imply it has a unique solution for every vector b. A matrix is invertible if its determinant is not equal to zero. We calculate the determinant of matrix a:

\[ \det(a) = (5)(2) - (-4)(4) = 10 + 16 = 26 \]

The determinant is not zero, which means the matrix a is invertible, and therefore the system ax = b has a solution for all b.

Answer 2

We can write the system in the following form:

`\left[\begin{array}{cccc}5&-4&4&2\end{array}\right] \left[\begin{array}{c}x_1\\x_2\\x_3\\x_4\end{array}\right] =b`
The above system is equivalent to the following equation:

`5x_1-4x_2+4x_3+2x_4=b,\forall b\in\mathbb{R}`
Of course, the above system has solution for any values of b since there is one equation and four variable, there is infinite number of solution each time.