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QuestionFind the matrix resulting from m×Hm \times H given the equations involving matrices.

Studdy Solution

STEP 1

Assumptions1. The given matrices are valid for matrix operations. . The scalar multiplication of matrices follows the distributive law.
3. The addition of matrices follows the commutative law.
4. The unknowns mm and HH are real numbers.

STEP 2

From the first equation, we can isolate mm by dividing both sides by the same matrix and the scalar/4.
m=4[−1248]4[−1248]m = \frac{4\left[\begin{array}{cc} -1 &2 \\ 4 &8\end{array}\right]}{\frac{}{4} \left[\begin{array}{cc} -1 &2 \\ 4 &8\end{array}\right]}

STEP 3

implify the right-hand side of the equation.
m=3=1×3=163m = \frac{}{\frac{3}{}} = \frac{}{1} \times \frac{}{3} = \frac{16}{3}

STEP 4

Now, let's solve for HH using the second equation. We can isolate HH by subtracting the same matrix from both sides.
H=[−23−1]−([14−2]+[32−6])H = \left[\begin{array}{lll} -2 &3 & -1\end{array}\right] - \left(\left[\begin{array}{lll} 1 &4 & -2\end{array}\right] + \left[\begin{array}{lll} 3 &2 & -6\end{array}\right]\right)

STEP 5

implify the right-hand side of the equation.
H=[−23−1]−[4−8]H = \left[\begin{array}{lll} -2 &3 & -1\end{array}\right] - \left[\begin{array}{lll} 4 & & -8\end{array}\right]

STEP 6

Perform the subtraction operation on the matrices.
H=[−2−43−6−1−(−8)]H = \left[\begin{array}{lll} -2-4 &3-6 & -1-(-8) \end{array}\right]

STEP 7

implify the right-hand side of the equation.
H=[−6−37]H = \left[\begin{array}{lll} -6 & -3 &7\end{array}\right]

STEP 8

Now that we have the values for mm and HH, we can find the result of m×Hm \times H.
m×H=163×[−6−37]m \times H = \frac{16}{3} \times \left[\begin{array}{lll} -6 & -3 &7\end{array}\right]

STEP 9

Perform the scalar multiplication operation on the matrix.
m×H=163×[−6−37]=[163×−6163×−3163×7]m \times H = \frac{16}{3} \times \left[\begin{array}{lll} -6 & -3 &7\end{array}\right] = \left[\begin{array}{lll} \frac{16}{3} \times -6 & \frac{16}{3} \times -3 & \frac{16}{3} \times7\end{array}\right]

STEP 10

implify the right-hand side of the equation.
m×H=[−32−161123]m \times H = \left[\begin{array}{lll} -32 & -16 & \frac{112}{3} \end{array}\right]The result of m×Hm \times H is the matrix [−32−161123]\left[\begin{array}{lll} -32 & -16 & \frac{112}{3} \end{array}\right].

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