Extended matrix

In linear algebra , an extended matrix is obtained by stringing together several given matrices, usually to perform the same elementary row operations on the matrices.

definition

With and as ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle A={\begin{bmatrix}1&3&2\\2&0&1\\5&2&2\end{bmatrix}},\quad B={\begin{bmatrix}4\\3\\1\end{bmatrix}}}$

the extended matrix is written as ${\displaystyle (A|B)}$

${\displaystyle (A|B)=\left[{\begin{array}{ccc|c}1&3&2&4\\2&0&1&3\\5&2&2&1\end{array}}\right].}$

Extended matrices are useful for solving systems of linear equations .

For a given number of unknowns, the number of solutions to the system of equations depends only on the rank of the matrix that represents the system of equations. According to Kronecker-Capelli's theorem , a system of linear equations in which the expanded matrix has a higher rank than the coefficient matrix has no solution; however, if the two matrices have the same rank, then at least one solution must exist. The solution is only unique if the rank and the number of variables are the same. Otherwise the solution has parameters, where is the difference between the number of variables and the rank, so in such cases there are infinitely many solutions. ${\displaystyle k}$ ${\displaystyle k}$

An extended matrix can also be used to find the inverse matrix by combining it with the identity matrix .

Find the inverse of a matrix

Let be the square 2 × 2 matrix ${\displaystyle C}$

${\displaystyle C={\begin{bmatrix}1&3\\-5&0\end{bmatrix}}}$.

To find the converse, one creates , where is the 2 × 2 identity matrix . The part of which belongs to is reduced to the identity matrix by applying only elementary row operations to : ${\displaystyle (C|I)}$ ${\displaystyle I}$ ${\displaystyle (C|I)}$ ${\displaystyle C}$ ${\displaystyle (C|I)}$

${\displaystyle (C|I)=\left[{\begin{array}{cc|cc}1&3&1&0\\-5&0&0&1\end{array}}\right]}$

${\displaystyle (I|C^{-1})=\left[{\begin{array}{cc|cc}1&0&0&-{\frac {1}{5}}\\0&1&{\frac {1}{3}}&{\frac {1}{15}}\end{array}}\right]}$

The right part is now the inverse of ${\displaystyle C}$

Existence and number of solutions

Consider the following system of linear equations

${\displaystyle {\displaystyle {\begin{aligned}x+y+2z&=3\\x+y+z&=1\\2x+2y+2z&=2.\end{aligned}}}}$

The coefficient matrix is

${\displaystyle A={\begin{bmatrix}1&1&2\\1&1&1\\2&2&2\\\end{bmatrix}},}$

and the expanded matrix is

${\displaystyle (A|B)=\left[{\begin{array}{ccc|c}1&1&2&3\\1&1&1&1\\2&2&2&2\end{array}}\right].}$

Since both have the same rank 2, there is at least one solution; and since the rank of both matrices is less than the number of variables, which is 3, there are infinitely many solutions.

In comparison, consider the following system of equations

${\displaystyle {\displaystyle {\begin{aligned}x+y+2z&=3\\x+y+z&=1\\2x+2y+2z&=5.\end{aligned}}}}$

The coefficient matrix is

${\displaystyle A={\begin{bmatrix}1&1&2\\1&1&1\\2&2&2\\\end{bmatrix}},}$

and the expanded matrix is

${\displaystyle (A|B)=\left[{\begin{array}{ccc|c}1&1&2&3\\1&1&1&1\\2&2&2&5\end{array}}\right].}$

In this example, the coefficient matrix is ​​rank 2 while the expanded matrix is ​​rank 3. So the system of equations has no solution. In fact, the increase in the linearly independent series has made the system of equations inconsistent .

literature

• A. Blickensdörfer-Ehlers, WG Eschmann, H Neunzert, K. Schelkes: Analysis 2: With an introduction to vector and matrix calculation A textbook and workbook . Springer, 1982, pp. 86-91
• Marvin Marcus and Henryk Minc: A survey of matrix theory and matrix inequalities. Dover Publications, 1992, ISBN 0-486-67102-X , p. 31

Web links

• Augmented Matrices at Paul's Online Notes (university script, English)
• Eric W. Weisstein : Augmented Matrix . In: MathWorld (English).