Linear Algebra Examples

$[\begin{array}{c} \frac{d}{a d - b c} & - \frac{b}{a d - b c} \\ - \frac{c}{a d - b c} & \frac{a}{a d - b d} \end{array}] [\begin{array}{c} - 6 & k \\ 0 & - 6 \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 1

Factor $d$ out of $a d - b d$ .

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Step 1.1

Factor $d$ out of $a d$ .

$[\begin{array}{c} \frac{d}{a d - b c} & - \frac{b}{a d - b c} \\ - \frac{c}{a d - b c} & \frac{a}{d a - b d} \end{array}] [\begin{array}{c} - 6 & k \\ 0 & - 6 \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 1.2

Factor $d$ out of $- b d$ .

$[\begin{array}{c} \frac{d}{a d - b c} & - \frac{b}{a d - b c} \\ - \frac{c}{a d - b c} & \frac{a}{d a + d (- b)} \end{array}] [\begin{array}{c} - 6 & k \\ 0 & - 6 \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 1.3

Factor $d$ out of $d a + d (- b)$ .

$[\begin{array}{c} \frac{d}{a d - b c} & - \frac{b}{a d - b c} \\ - \frac{c}{a d - b c} & \frac{a}{d (a - b)} \end{array}] [\begin{array}{c} - 6 & k \\ 0 & - 6 \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 2

Multiply $[\begin{array}{c} \frac{d}{a d - b c} & - \frac{b}{a d - b c} \\ - \frac{c}{a d - b c} & \frac{a}{d (a - b)} \end{array}] [\begin{array}{c} - 6 & k \\ 0 & - 6 \end{array}]$ .

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Step 2.1

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 $2 \times 2$ and the second matrix is $2 \times 2$ .

Step 2.2

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

$[\begin{array}{c} \frac{d}{a d - b c} \cdot - 6 - \frac{b}{a d - b c} \cdot 0 & \frac{d}{a d - b c} k - \frac{b}{a d - b c} \cdot - 6 \\ - \frac{c}{a d - b c} \cdot - 6 + \frac{a}{d (a - b)} \cdot 0 & - \frac{c}{a d - b c} k + \frac{a}{d (a - b)} \cdot - 6 \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 2.3

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

$[\begin{array}{c} - \frac{6 d}{a d - b c} & \frac{d k + 6 b}{a d - b c} \\ \frac{6 c}{a d - b c} & - \frac{k c d a - k c d b + 6 a^{2} d - 6 a b c}{d (a - b) (a d - b c)} \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3

Multiply $[\begin{array}{c} - \frac{6 d}{a d - b c} & \frac{d k + 6 b}{a d - b c} \\ \frac{6 c}{a d - b c} & - \frac{k c d a - k c d b + 6 a^{2} d - 6 a b c}{d (a - b) (a d - b c)} \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}]$ .

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Step 3.1

Step 3.2

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

$[\begin{array}{c} - \frac{6 d}{a d - b c} a + \frac{d k + 6 b}{a d - b c} c & - \frac{6 d}{a d - b c} b + \frac{d k + 6 b}{a d - b c} d \\ \frac{6 c}{a d - b c} a - \frac{k c d a - k c d b + 6 a^{2} d - 6 a b c}{d (a - b) (a d - b c)} c & \frac{6 c}{a d - b c} b - \frac{k c d a - k c d b + 6 a^{2} d - 6 a b c}{d (a - b) (a d - b c)} d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3.3

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

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Step 3.3.1

Apply the distributive property.

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a + 6 c b (- b) - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3.3.2

Multiply $b$ by $b$ by adding the exponents.

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Step 3.3.2.1

Move $b$ .

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a + 6 c (b \cdot b) \cdot - 1 - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3.3.2.2

Multiply $b$ by $b$ .

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a + 6 c b^{2} \cdot - 1 - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3.3.3

Multiply $- 1$ by $6$ .

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3.3.4

Apply the distributive property.

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - (k c d a) - (- k c d b) - (6 a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3.3.5

Simplify.

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Step 3.3.5.1

Multiply $- (- k c d b)$ .

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Step 3.3.5.1.1

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + 1 (k c d b) - (6 a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3.3.5.1.2

Multiply $k$ by $1$ .

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k (c d b) - (6 a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3.3.5.2

Multiply $6$ by $- 1$ .

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k (c d b) - 6 (a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3.3.5.3

Multiply $- 6$ by $- 1$ .

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k (c d b) - 6 (a^{2} d) + 6 (a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3.3.6

Remove parentheses.

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d + 6 a b c}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3.3.7

Add $6 c b a$ and $6 a b c$ .

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Step 3.3.7.1

Move $b$ .

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 a b c + 6 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 3.3.7.2

Add $6 a b c$ and $6 a b c$ .

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{12 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 4

Write as a linear system of equations.

$- \frac{6 a d - c d k - 6 c b}{a d - b c} = 1$

$\frac{d^{2} k}{a d - b c} = 0$

$- \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} = 0$

$\frac{12 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} = 1$

Step 5

Reduce the system.

$- \frac{6 a d - c d k - 6 c b}{a d - b c} = 1, \frac{d^{2} k}{a d - b c} = 0, - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} = 0, \frac{12 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} = 1$