Linear Algebra Examples

$[\begin{array}{c} 1 & 0 & 2 \\ 2 & 1 & 1 \\ 6 & 4 & 1 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 1 \\ - 2 \\ 3 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 1 & 0 & 2 \\ 2 & 1 & 1 \\ 6 & 4 & 1 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}]$ .

Tap for more steps...

Step 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $3 \times 3$ and the second matrix is $3 \times 1$ .

Step 1.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 1 x + 0 y + 2 z \\ 2 x + 1 y + 1 z \\ 6 x + 4 y + 1 z \end{array}] = [\begin{array}{c} 1 \\ - 2 \\ 3 \end{array}]$

Step 1.3

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} x + 2 z \\ 2 x + y + z \\ 6 x + 4 y + z \end{array}] = [\begin{array}{c} 1 \\ - 2 \\ 3 \end{array}]$

Step 2

Write as a linear system of equations.

$x + 2 z = 1$

$2 x + y + z = - 2$

$6 x + 4 y + z = 3$

Step 3

Solve the system of equations.

Tap for more steps...

Step 3.1

Subtract $2 z$ from both sides of the equation.

$x = 1 - 2 z$

$2 x + y + z = - 2$

$6 x + 4 y + z = 3$

Step 3.2

Replace all occurrences of $x$ with $1 - 2 z$ in each equation.

Tap for more steps...

Step 3.2.1

Replace all occurrences of $x$ in $2 x + y + z = - 2$ with $1 - 2 z$ .

$2 (1 - 2 z) + y + z = - 2$

$x = 1 - 2 z$

$6 x + 4 y + z = 3$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Simplify $2 (1 - 2 z) + y + z$ .

Tap for more steps...

Step 3.2.2.1.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1.1

Apply the distributive property.

$2 \cdot 1 + 2 (- 2 z) + y + z = - 2$

$x = 1 - 2 z$

$6 x + 4 y + z = 3$

Step 3.2.2.1.1.2

Multiply $2$ by $1$ .

$2 + 2 (- 2 z) + y + z = - 2$

$x = 1 - 2 z$

$6 x + 4 y + z = 3$

Step 3.2.2.1.1.3

Multiply $- 2$ by $2$ .

$2 - 4 z + y + z = - 2$

$x = 1 - 2 z$

$6 x + 4 y + z = 3$

$2 - 4 z + y + z = - 2$

$x = 1 - 2 z$

$6 x + 4 y + z = 3$

Step 3.2.2.1.2

Add $- 4 z$ and $z$ .

$2 + y - 3 z = - 2$

$x = 1 - 2 z$

$6 x + 4 y + z = 3$

$2 + y - 3 z = - 2$

$x = 1 - 2 z$

$6 x + 4 y + z = 3$

$2 + y - 3 z = - 2$

$x = 1 - 2 z$

$6 x + 4 y + z = 3$

Step 3.2.3

Replace all occurrences of $x$ in $6 x + 4 y + z = 3$ with $1 - 2 z$ .

$6 (1 - 2 z) + 4 y + z = 3$

$2 + y - 3 z = - 2$

$x = 1 - 2 z$

Step 3.2.4

Simplify the left side.

Tap for more steps...

Step 3.2.4.1

Simplify $6 (1 - 2 z) + 4 y + z$ .

Tap for more steps...

Step 3.2.4.1.1

Simplify each term.

Tap for more steps...

Step 3.2.4.1.1.1

Apply the distributive property.

$6 \cdot 1 + 6 (- 2 z) + 4 y + z = 3$

$2 + y - 3 z = - 2$

$x = 1 - 2 z$

Step 3.2.4.1.1.2

Multiply $6$ by $1$ .

$6 + 6 (- 2 z) + 4 y + z = 3$

$2 + y - 3 z = - 2$

$x = 1 - 2 z$

Step 3.2.4.1.1.3

Multiply $- 2$ by $6$ .

$6 - 12 z + 4 y + z = 3$

$2 + y - 3 z = - 2$

$x = 1 - 2 z$

$6 - 12 z + 4 y + z = 3$

$2 + y - 3 z = - 2$

$x = 1 - 2 z$

Step 3.2.4.1.2

Add $- 12 z$ and $z$ .

$6 + 4 y - 11 z = 3$

$2 + y - 3 z = - 2$

$x = 1 - 2 z$

$6 + 4 y - 11 z = 3$

$2 + y - 3 z = - 2$

$x = 1 - 2 z$

$6 + 4 y - 11 z = 3$

$2 + y - 3 z = - 2$

$x = 1 - 2 z$

$6 + 4 y - 11 z = 3$

$2 + y - 3 z = - 2$

$x = 1 - 2 z$

Step 3.3

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 3.3.1

Subtract $2$ from both sides of the equation.

$y - 3 z = - 2 - 2$

$6 + 4 y - 11 z = 3$

$x = 1 - 2 z$

Step 3.3.2

Add $3 z$ to both sides of the equation.

$y = - 2 - 2 + 3 z$

$6 + 4 y - 11 z = 3$

$x = 1 - 2 z$

Step 3.3.3

Subtract $2$ from $- 2$ .

$y = - 4 + 3 z$

$6 + 4 y - 11 z = 3$

$x = 1 - 2 z$

$y = - 4 + 3 z$

$6 + 4 y - 11 z = 3$

$x = 1 - 2 z$

Step 3.4

Replace all occurrences of $y$ with $- 4 + 3 z$ in each equation.

Tap for more steps...

Step 3.4.1

Replace all occurrences of $y$ in $6 + 4 y - 11 z = 3$ with $- 4 + 3 z$ .

$6 + 4 (- 4 + 3 z) - 11 z = 3$

$y = - 4 + 3 z$

$x = 1 - 2 z$

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Simplify $6 + 4 (- 4 + 3 z) - 11 z$ .

Tap for more steps...

Step 3.4.2.1.1

Simplify each term.

Tap for more steps...

Step 3.4.2.1.1.1

Apply the distributive property.

$6 + 4 \cdot - 4 + 4 (3 z) - 11 z = 3$

$y = - 4 + 3 z$

$x = 1 - 2 z$

Step 3.4.2.1.1.2

Multiply $4$ by $- 4$ .

$6 - 16 + 4 (3 z) - 11 z = 3$

$y = - 4 + 3 z$

$x = 1 - 2 z$

Step 3.4.2.1.1.3

Multiply $3$ by $4$ .

$6 - 16 + 12 z - 11 z = 3$

$y = - 4 + 3 z$

$x = 1 - 2 z$

$6 - 16 + 12 z - 11 z = 3$

$y = - 4 + 3 z$

$x = 1 - 2 z$

Step 3.4.2.1.2

Simplify by adding terms.

Tap for more steps...

Step 3.4.2.1.2.1

Subtract $16$ from $6$ .

$- 10 + 12 z - 11 z = 3$

$y = - 4 + 3 z$

$x = 1 - 2 z$

Step 3.4.2.1.2.2

Subtract $11 z$ from $12 z$ .

$- 10 + z = 3$

$y = - 4 + 3 z$

$x = 1 - 2 z$

$- 10 + z = 3$

$y = - 4 + 3 z$

$x = 1 - 2 z$

$- 10 + z = 3$

$y = - 4 + 3 z$

$x = 1 - 2 z$

$- 10 + z = 3$

$y = - 4 + 3 z$

$x = 1 - 2 z$

$- 10 + z = 3$

$y = - 4 + 3 z$

$x = 1 - 2 z$

Step 3.5

Move all terms not containing $z$ to the right side of the equation.

Tap for more steps...

Step 3.5.1

Add $10$ to both sides of the equation.

$z = 3 + 10$

$y = - 4 + 3 z$

$x = 1 - 2 z$

Step 3.5.2

Add $3$ and $10$ .

$z = 13$

$y = - 4 + 3 z$

$x = 1 - 2 z$

$z = 13$

$y = - 4 + 3 z$

$x = 1 - 2 z$

Step 3.6

Replace all occurrences of $z$ with $13$ in each equation.

Tap for more steps...

Step 3.6.1

Replace all occurrences of $z$ in $y = - 4 + 3 z$ with $13$ .

$y = - 4 + 3 (13)$

$z = 13$

$x = 1 - 2 z$

Step 3.6.2

Simplify the right side.

Tap for more steps...

Step 3.6.2.1

Simplify $- 4 + 3 (13)$ .

Tap for more steps...

Step 3.6.2.1.1

Multiply $3$ by $13$ .

$y = - 4 + 39$

$z = 13$

$x = 1 - 2 z$

Step 3.6.2.1.2

Add $- 4$ and $39$ .

$y = 35$

$z = 13$

$x = 1 - 2 z$

$y = 35$

$z = 13$

$x = 1 - 2 z$

$y = 35$

$z = 13$

$x = 1 - 2 z$

Step 3.6.3

Replace all occurrences of $z$ in $x = 1 - 2 z$ with $13$ .

$x = 1 - 2 \cdot 13$

$y = 35$

$z = 13$

Step 3.6.4

Simplify the right side.

Tap for more steps...

Step 3.6.4.1

Simplify $1 - 2 \cdot 13$ .

Tap for more steps...

Step 3.6.4.1.1

Multiply $- 2$ by $13$ .

$x = 1 - 26$

$y = 35$

$z = 13$

Step 3.6.4.1.2

Subtract $26$ from $1$ .

$x = - 25$

$y = 35$

$z = 13$

$x = - 25$

$y = 35$

$z = 13$

$x = - 25$

$y = 35$

$z = 13$

$x = - 25$

$y = 35$

$z = 13$

Step 3.7

List all of the solutions.

$x = - 25, y = 35, z = 13$