Matrices and Determinants Part 4

Concept: The problem requires determining the coefficients of a linear combination of the standard basis vectors, which is achieved by solving a system of linear equations derived from matrix multiplication.Formula/Principle: Matrix multiplication: If $M$ is a matrix and $v$ is a vector, the resulting vector is found by multiplying the matrix by the vector.Solution/Analysis:1. Determine the Matrix M: Since the columns of a matrix $M$ are the results of multiplying $M$ by the standard basis vectors $\begin{pmatrix}1\\0\\0\end{pmatrix}$, $\begin{pmatrix}0\\1\\0\end{pmatrix}$, and $\begin{pmatrix}0\\0\\1\end{pmatrix}$, we can construct $M$:$M = \begin{pmatrix}1 & 0 & -1\\2 & 1 & 1\\3 & 2 & 1\end{pmatrix}$2. Set up the System of Equations: We are given that $M \begin{pmatrix}x \\ y \\ z\end{pmatrix} = \begin{pmatrix}1 \\ 7 \\ 11\end{pmatrix}$. Performing the matrix multiplication yields the following system of linear equations:$1x + 0y - 1z \;\; = 1 \;\; (1)$$2x + 1y + 1z \;\; = 7 \;\; (2)$$3x + 2y + 1z \;\; = 11 \;\; (3)$3. Solve the System:

• From equation (1), we get: $x - z = 1$, which implies $x = 1 + z$.
• Substitute $x$ into equation (2):
  $2(1 + z) + y + z = 7$
  $2 + 2z + y + z = 7$
  $y + 3z = 5 \;\; (4)$
• Substitute $x$ into equation (3):
  $3(1 + z) + 2y + z = 11$
  $3 + 3z + 2y + z = 11$
  $2y + 4z = 8$
  Dividing by 2, we get: $y + 2z = 4 \;\; (5)$

4. Solve for y and z: Subtract equation (5) from equation (4):$(y + 3z) - (y + 2z) = 5 - 4$$z = 1$Substitute $z=1$ back into equation (5):$y + 2(1) = 4$$y = 2$5. Solve for x: Use the relation $x = 1 + z$:$x = 1 + 1$$x = 2$6. Calculate the Final Value: The question asks for the value of $x + y + z$:$x + y + z = 2 + 2 + 1 = 5$