Answered

Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

The matrix equation represents a system of equations.

[tex]\[
\left[\begin{array}{ll}
2 & 7 \\
2 & 6
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right]=\left[\begin{array}{l}
4 \\
6
\end{array}\right]
\][/tex]

Solve for [tex]\( y \)[/tex] using matrices. Show or explain all necessary steps.

Sagot :

GinnyAnswer
To solve the matrix equation

[tex]\[ \left[\begin{array}{cc} 2 & 7 \\ 2 & 6 \end{array}\right]\left[\begin{array}{c} x \\ y \end{array}\right]=\left[\begin{array}{c} 4 \\ 6 \end{array}\right], \][/tex]

we will use the inverse of the coefficient matrix.

### Step 1: Define the matrices

Let [tex]\( A \)[/tex] be the coefficient matrix, [tex]\( \mathbf{x} \)[/tex] be the variable vector, and [tex]\( \mathbf{b} \)[/tex] be the constant vector.

[tex]\[ A = \left[\begin{array}{cc} 2 & 7 \\ 2 & 6 \end{array}\right], \quad \mathbf{x} = \left[\begin{array}{c} x \\ y \end{array}\right], \quad \mathbf{b} = \left[\begin{array}{c} 4 \\ 6 \end{array}\right] \][/tex]

### Step 2: Calculate the inverse of matrix [tex]\( A \)[/tex]

We need to find [tex]\( A^{-1} \)[/tex], the inverse of matrix [tex]\( A \)[/tex]. Recall that if [tex]\( A \)[/tex] is a 2x2 matrix

[tex]\[ A = \left[\begin{array}{cc} a & b \\ c & d \end{array}\right], \][/tex]

then the inverse [tex]\( A^{-1} \)[/tex] is given by

[tex]\[ A^{-1} = \frac{1}{ad - bc} \left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right] \][/tex]

For our matrix [tex]\( A \)[/tex]:

[tex]\[ a = 2, \quad b = 7, \quad c = 2, \quad d = 6 \][/tex]

First, calculate the determinant [tex]\( \text{det}(A) \)[/tex]:

[tex]\[ \text{det}(A) = ad - bc = (2)(6) - (2)(7) = 12 - 14 = -2 \][/tex]

Now, calculate [tex]\( A^{-1} \)[/tex]:

[tex]\[ A^{-1} = \frac{1}{-2} \left[\begin{array}{cc} 6 & -7 \\ -2 & 2 \end{array}\right] = \left[\begin{array}{cc} -3 & \frac{7}{2} \\ 1 & -1 \end{array}\right] = \left[\begin{array}{cc} -3 & 3.5 \\ 1 & -1 \end{array}\right] \][/tex]

### Step 3: Multiply [tex]\( A^{-1} \)[/tex] by [tex]\( \mathbf{b} \)[/tex]

To find the vector [tex]\( \mathbf{x} \)[/tex], multiply [tex]\( A^{-1} \)[/tex] by [tex]\( \mathbf{b} \)[/tex]:

[tex]\[ \mathbf{x} = A^{-1} \mathbf{b} \][/tex]

Perform the matrix multiplication:

[tex]\[ \mathbf{x} = \left[\begin{array}{cc} -3 & 3.5 \\ 1 & -1 \end{array}\right] \left[\begin{array}{c} 4 \\ 6 \end{array}\right] \][/tex]

Calculate each component of [tex]\( \mathbf{x} \)[/tex]:

1. For [tex]\( x \)[/tex]:

[tex]\[ x = (-3)(4) + (3.5)(6) = -12 + 21 = 9 \][/tex]

2. For [tex]\( y \)[/tex]:

[tex]\[ y = (1)(4) + (-1)(6) = 4 - 6 = -2 \][/tex]

### Step 4: Verify the solution [tex]\( x = 9 \)[/tex] and [tex]\( y = -2 \)[/tex]

Thus, the solution to the system of equations is:

[tex]\[ x = 9, \quad y = -2 \][/tex]

So, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the given matrix equation are [tex]\( x = 9 \)[/tex] and [tex]\( y = -2 \)[/tex]. The inverse matrix [tex]\( A^{-1} \)[/tex] is

[tex]\[ \left[\begin{array}{cc} -3 & 3.5 \\ 1 & -1 \end{array}\right] \][/tex]

and the result calculated using the inverse is:

[tex]\[ \begin{pmatrix} 9 \\ -2 \end{pmatrix}. \][/tex]
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.