Math / Algebra

QuestionSolve the matrix equation $2 \mathrm{X}+\mathrm{A}=\mathrm{B}$ for X if $\mathrm{A}=\left[\begin{array}{rr}-4 & -4 \\ 8 & 7 \\ -3 & 4\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{rr}-4 & -2 \\ 8 & -9 \\ 5 & -2\end{array}\right]$ . $\mathrm{X}=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

What is this asking? We need to solve for the unknown matrix $X$ in the given matrix equation, just like solving for $x$ in a regular equation! Watch out! Matrix operations have specific rules, so be careful when adding, subtracting, and multiplying matrices.
Remember, we can't just divide matrices!

STEP 2

1. Isolate the X term
2. Multiply by the scalar
3. Calculate the result

STEP 3

We start with the given equation: $2X + A = B$

STEP 4

To isolate the term with $X$ , we subtract $A$ from both sides of the equation: $2X + A - A = B - A$ $2X = B - A$ This is just like solving a regular equation!

STEP 5

Now, we want to get $X$ all by itself.
Since $X$ is being multiplied by 2, we multiply both sides of the equation by 1/2: $\frac{1}{2} \cdot 2X = \frac{1}{2}(B - A)$ This simplifies to: $X = \frac{1}{2}(B - A)$ So we're almost there!

STEP 6

Now, let's substitute the given matrices $A$ and $B$ into our equation: $X = \frac{1}{2} \left( \begin{array}{rr} -4 & -2 \\ 8 & -9 \\ 5 & -2 \end{array} \right) - \frac{1}{2} \left( \begin{array}{rr} -4 & -4 \\ 8 & 7 \\ -3 & 4 \end{array} \right)$

STEP 7

Remember, when subtracting matrices, we subtract corresponding elements: $X = \frac{1}{2}\left[\begin{array}{rr}0 & 2 \\ 0 & -16 \\ 8 & -6\end{array}\right]$ $X = \frac{1}{2}\begin{array}{rr}0 & 2 \\ 0 & -16 \\ 8 & -6\end{array}$

STEP 8

Finally, we multiply each element of the resulting matrix by $\frac{1}{2}$ : $X = \begin{array}{cc} 0 & 1 \\ 0 & -8 \\ 4 & -3 \end{array}$ $X = \begin{array}{cc} 0 & 1 \\ 0 & -8 \\ 4 & -3 \end{array}$

STEP 9

So, our solution for the matrix $X$ is: $X = \begin{array}{cc} 0 & 1 \\ 0 & -8 \\ 4 & -3 \end{array}$

Was this helpful?

QuestionSolve the matrix equation $2 \mathrm{X}+\mathrm{A}=\mathrm{B}$ for X if $\mathrm{A}=\left[\begin{array}{rr}-4 & -4 \\ 8 & 7 \\ -3 & 4\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{rr}-4 & -2 \\ 8 & -9 \\ 5 & -2\end{array}\right]$ . $\mathrm{X}=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

What is this asking? We need to solve for the unknown matrix $X$ in the given matrix equation, just like solving for $x$ in a regular equation! Watch out! Matrix operations have specific rules, so be careful when adding, subtracting, and multiplying matrices.
Remember, we can't just divide matrices!

STEP 2

1. Isolate the X term
2. Multiply by the scalar
3. Calculate the result

STEP 3

We start with the given equation: $2X + A = B$

STEP 4

To isolate the term with $X$ , we subtract $A$ from both sides of the equation: $2X + A - A = B - A$ $2X = B - A$ This is just like solving a regular equation!

STEP 5

Now, we want to get $X$ all by itself.
Since $X$ is being multiplied by 2, we multiply both sides of the equation by 1/2: $\frac{1}{2} \cdot 2X = \frac{1}{2}(B - A)$ This simplifies to: $X = \frac{1}{2}(B - A)$ So we're almost there!

STEP 6

Now, let's substitute the given matrices $A$ and $B$ into our equation: $X = \frac{1}{2} \left( \begin{array}{rr} -4 & -2 \\ 8 & -9 \\ 5 & -2 \end{array} \right) - \frac{1}{2} \left( \begin{array}{rr} -4 & -4 \\ 8 & 7 \\ -3 & 4 \end{array} \right)$

STEP 7

Remember, when subtracting matrices, we subtract corresponding elements: $X = \frac{1}{2}\left[\begin{array}{rr}0 & 2 \\ 0 & -16 \\ 8 & -6\end{array}\right]$ $X = \frac{1}{2}\begin{array}{rr}0 & 2 \\ 0 & -16 \\ 8 & -6\end{array}$

STEP 8

Finally, we multiply each element of the resulting matrix by $\frac{1}{2}$ : $X = \begin{array}{cc} 0 & 1 \\ 0 & -8 \\ 4 & -3 \end{array}$ $X = \begin{array}{cc} 0 & 1 \\ 0 & -8 \\ 4 & -3 \end{array}$

STEP 9

So, our solution for the matrix $X$ is: $X = \begin{array}{cc} 0 & 1 \\ 0 & -8 \\ 4 & -3 \end{array}$

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

What is this asking? We need to solve for the unknown matrix XXX in the given matrix equation, just like solving for xxx in a regular equation! Watch out! Matrix operations have specific rules, so be careful when adding, subtracting, and multiplying matrices.Remember, we can't just divide matrices!

STEP 2

1. Isolate the X term2. Multiply by the scalar3. Calculate the result

STEP 3

We start with the given equation: 2X+A=B2X + A = B2X+A=B

STEP 4

To isolate the term with XXX, we subtract AAA from both sides of the equation: 2X+A−A=B−A2X + A - A = B - A2X+A−A=B−A 2X=B−A2X = B - A2X=B−AThis is just like solving a regular equation!

STEP 5

STEP 6

STEP 7

STEP 8

Finally, we multiply each element of the resulting matrix by 12\frac{1}{2}21​: X=010−84−3X = \begin{array}{cc} 0 & 1 \\ 0 & -8 \\ 4 & -3 \end{array}X=004​1−8−3​ X=010−84−3 X = \begin{array}{cc} 0 & 1 \\ 0 & -8 \\ 4 & -3 \end{array} X=004​1−8−3​

STEP 9

So, our solution for the matrix XXX is: X=010−84−3 X = \begin{array}{cc} 0 & 1 \\ 0 & -8 \\ 4 & -3 \end{array} X=004​1−8−3​

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Isolate the X term
2. Multiply by the scalar
3. Calculate the result

We start with the given equation: $2X + A = B$

To isolate the term with $X$ , we subtract $A$ from both sides of the equation: $2X + A - A = B - A$ $2X = B - A$ This is just like solving a regular equation!

Finally, we multiply each element of the resulting matrix by $\frac{1}{2}$ : $X = \begin{array}{cc} 0 & 1 \\ 0 & -8 \\ 4 & -3 \end{array}$ $X = \begin{array}{cc} 0 & 1 \\ 0 & -8 \\ 4 & -3 \end{array}$

So, our solution for the matrix $X$ is: $X = \begin{array}{cc} 0 & 1 \\ 0 & -8 \\ 4 & -3 \end{array}$