Algebra - Matrices

2. Equivalent matrices


  • Elementary row/column operations:
    There are six transformations (operations) on a matrix, out of which three are on rows and other three are on columns of a matrix.
    (a) Swap rows or columns: Interchange of any two rows or two columns.
    Notation: $R_{i}\leftrightarrow R_{j}$ for interchanging $i^{th}$ row to $j^{th}$ row and $C_{i}\leftrightarrow C_{j}$ for interchanging $i^{th}$ column to $j^{th}$ column.
    (b) Multiplication of the elements of any row or column by a scalar(non-zero).
    Notation: $R_{i}\rightarrow kR_{i}$ for multiplication of each element of $i^{th}$ row by $k$, where $k\neq 0$ and $C_{i}\rightarrow kC_{i}$ for multiplication of each element of $i^{th}$ column by $k$, where $k\neq 0$.
    (c) Addition to the elements of any row or columns, the corresponding elements of any other row or column multiplied by any non zero scalar.
    Notation: $R_{i}\rightarrow R_{i}+kR_{j}$ for the addition of the elements of $i^{th}$ row, the corresponding elements of $j^{th}$ row multiplied by $k$ ($k\neq 0$) and $C_{i}\rightarrow C_{i}+kC_{j}$ for the addition of the elements of $i^{th}$ column, the corresponding elements of $j^{th}$ column multiplied by $k$ ($k\neq 0$).

  • Definition 2.1: Two matrices $A$ and $B$ are said to be equivalent if the matrix $B$ can be obtained from the matrix $A$ by applying finite sqeuence of elementary row/column operations denoted by $A\sim B$ or $A \approxeq B$.

  • Definition 2.1: Two matrices $A$ and $B$ are said to be row equivalent if the matrix $B$ can be obtained from the matrix $A$ by applying finite sqeuence of elementary row operations.

  • The equivalent relation is an equivalence realtion i.e., equivalent relation is reflexive ($A \approxeq A$), symmetric ($A \approxeq B \Rightarrow B \approxeq A$) and transitive ($A \approxeq B$ and $B \approxeq C$ $\Rightarrow A \approxeq C$)

  • Problem 2.1: Show that the row equivalence of the matrix $\begin{bmatrix} 1&2&-2&3\\2&5&4&7\\-1&-3&2&-1\\2&4&-1&3 \end{bmatrix}$ is $\begin{bmatrix} 1&2&-2&3\\0&1&8&1\\0&0&1&3/8\\0&0&0&1 \end{bmatrix}$
    Solution: Consider $\begin{bmatrix} 1&2&-2&3\\2&5&4&7\\-1&-3&2&-1\\2&4&-1&3 \end{bmatrix}$
    $R_2\rightarrow R_2-2R_1$, $R_3\rightarrow R_3+R_1$, $R_4\rightarrow R_4-2R_1$
    $\approxeq\begin{bmatrix} 1&2&-2&3\\0&1&8&1\\0&-1&0&2\\0&0&3&-3 \end{bmatrix}$
    $R_3\rightarrow R_3+R_2$
    $\approxeq\begin{bmatrix} 1&2&-2&3\\0&1&8&1\\0&0&8&3\\0&0&3&-3 \end{bmatrix}$
    $R_3\rightarrow R_3/8$
    $\approxeq\begin{bmatrix} 1&2&-2&3\\0&1&8&1\\0&0&1&3/8\\0&0&3&-3 \end{bmatrix}$
    $R_4\rightarrow R_4-3R_3$
    $\approxeq\begin{bmatrix} 1&2&-2&3\\0&1&8&1\\0&0&1&3/8\\0&0&0&-33/8 \end{bmatrix}$
    $R_4\rightarrow -8R_4/33$
    $\approxeq\begin{bmatrix} 1&2&-2&3\\0&1&8&1\\0&0&1&3/8\\0&0&0&1 \end{bmatrix}$ $\spadesuit$

  • Problem 2.2: Show that row equivalence of the matrix $\begin{bmatrix} 1&1&2\\-2&-1&-3\\5&2&7 \end{bmatrix}$ is $\begin{bmatrix} 1&0&1\\0&1&1\\0&0&0 \end{bmatrix}$
    Solution: Consider $\begin{bmatrix} 1&1&2\\-2&-1&-3\\5&2&7 \end{bmatrix}$
    $R_2\rightarrow R_2+2R_1$, $R_3\rightarrow R_3-5R_1$
    $\approxeq\begin{bmatrix} 1&1&2\\0&1&1\\0&-3&-3 \end{bmatrix}$
    $R_3\rightarrow R_3+3R_2$
    $\approxeq\begin{bmatrix} 1&1&2\\0&1&1\\0&0&0 \end{bmatrix}$
    $R_1\rightarrow R_1-R_2$
    $\approxeq\begin{bmatrix} 1&0&1\\0&1&1\\0&0&0 \end{bmatrix}$ $\spadesuit$

  • Problem 2.3: Show the row equivalence of $\begin{bmatrix} 2&-4&0\\1&2&1\\-1&0&3 \end{bmatrix}$ is $\begin{bmatrix} 1&2&1\\0&1&2\\0&0&1 \end{bmatrix}$
    Solution: Consider $\begin{bmatrix} 2&-4&0\\1&2&1\\-1&0&3 \end{bmatrix}$
    $R_1\leftrightarrow R_2$
    $\approxeq\begin{bmatrix} 1&2&1\\2&-4&0\\-1&0&3 \end{bmatrix}$
    $R_2\rightarrow R_2-2R_1$, $R_3\rightarrow R_3+R_1$
    $\approxeq \begin{bmatrix} 1&2&1\\0&-8&-2\\0&2&4 \end{bmatrix}$
    $R_2\leftrightarrow R_3$
    $\approxeq \begin{bmatrix} 1&2&1\\0&2&4\\0&-8&-2 \end{bmatrix}$
    $R_2\rightarrow R_2/2$
    $\approxeq \begin{bmatrix} 1&2&1\\0&1&2\\0&-8&-2 \end{bmatrix}$
    $R_3\rightarrow R_3+8R_1$
    $\approxeq \begin{bmatrix} 1&2&1\\0&1&2\\0&0&14 \end{bmatrix}$
    $R_3\rightarrow R_3/14$
    $\approxeq \begin{bmatrix} 1&2&1\\0&1&2\\0&0&1 \end{bmatrix}$ $\spadesuit$

  • Exercise 2.1: Show that row equivalence form of $\begin{bmatrix} 0&0&1&3\\2&4&0&-8\\1&2&1&-1 \end{bmatrix}$ is $\begin{bmatrix} 1&2&1&-1\\0&0&1&3\\0&0&0&0 \end{bmatrix}$

  • Exercise 2.2: Show that the matrices are equivalent $A=\begin{bmatrix} 1&-1&0\\2&1&1 \end{bmatrix}$ and $B=\begin{bmatrix} 3&0&1\\0&3&1 \end{bmatrix}$

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