Math

QuestionFind the values of hh for which the matrix is consistent:
[1235h15] \left[\begin{array}{rrr} 1 & 2 & -3 \\ 5 & h & -15 \end{array}\right]
Options: A. hh \neq B. h=h= C. Consistent for all hh D. Not consistent for any hh.

Studdy Solution

STEP 1

Assumptions1. The given matrix is an augmented matrix of a linear system. . We are asked to find the value(s) of h for which the system is consistent.
3. A system is consistent if there are either one or infinitely many solutions.

STEP 2

The given matrix is[125h15]\left[\begin{array}{rrr} 1 &2 & - \\ 5 & h & -15\end{array}\right]

STEP 3

We can write the augmented matrix as a system of linear equations. The first row represents the equation x+2y=3x +2y = -3 and the second row represents the equation 5x+hy=155x + hy = -15.

STEP 4

For the system to be consistent, there should be no contradiction between the two equations. A contradiction would occur if the left side of the second equation is a multiple of the left side of the first equation, but the right side is not a multiple of the right side of the first equation.

STEP 5

The left side of the second equation is a multiple of the left side of the first equation if 5x+hy=5(x+2y)5x + hy =5(x +2y), which simplifies to h=10h =10.

STEP 6

The right side of the second equation is a multiple of the right side of the first equation if 15=5(3)-15 =5(-3), which is true for all values of h.

STEP 7

Therefore, the system is consistent if h=10h =10.
So, the correct choice is B. The matrix is the augmented matrix of a consistent linear system if h=10h =10.

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