§ 15. Elementary Matrix Operations and Elementary Matrices¶
Elementary Matrix Operations and Elementary Matrices¶
Definition 15.1 : Elementary Row and Column Operations
Let \(A\) be an \(m \times n\) matrix. Any one of the following three operations on the rows [columns] of \(A\) is called an elementary row [column] operation:
- (1) interchanging any two rows [columns] of \(A\);
- (2) multiplying any row [column] of \(A\) by a nonzero scalar;
- (3) adding any scalar multiple of a row [column] of \(A\) to another row [column].
Any of these three operations is called an elementary operation. Elementary operations are of type 1, type 2, or type 3 depending on whether they are obtained by (1), (2), or (3).
Definition 15.2 : Elementary Matrices
An \(n \times n\) elementary matrix is a matrix obtained by performing an elementary operation on \(I_{n}\). The elementary matrix is said to be of type 1, 2, or 3 according to whether the elementary operation performed on \(I_{n}\) is a type 1, 2, or 3 operation, respectively.
Theorem 15.3 : Elementary operations can be represented as multiplication with elementary matrices
Let \(A \in \mathrm{M}_{m \times n}(F)\), and suppose that \(B\) is obtained from \(A\) by performing an elementary row [column] operation. Then there exists an \(m \times m[n \times n]\) elementary matrix \(E\) such that \(B=E A[B=A E]\). In fact, \(E\) is obtained from \(I_{m}\left[I_{n}\right]\) by performing the same elementary row [column] operation as that which was performed on \(A\) to obtain \(B\). Conversely, if \(E\) is an elementary \(m \times m[n \times n]\) matrix, then \(E A[A E]\) is the matrix obtained from \(A\) by performing the same elementary row [column] operation as that which produces \(E\) from \(I_{m}\left[I_{n}\right]\).
Proof
We prove the row version; the column version is analogous.
Suppose that \(B\) is obtained from \(A\) by an elementary row operation. Let \(E\) be the matrix obtained from \(I_m\) by performing that same row operation. We claim that \(B=EA\).
Let the rows of \(A\) be \(r_1,\ldots,r_m \in F^{1\times n}\). Since left-multiplication by a matrix forms linear combinations of the rows of the right factor, the \(i\)th row of \(EA\) is a linear combination of the rows of \(A\) with coefficients taken from the \(i\)th row of \(E\).
We now check this in the three types of elementary row operations.
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(1) Interchanging rows \(i\) and \(j\). In this case, \(E\) is obtained from \(I_m\) by swapping rows \(i\) and \(j\), so \(E\) is the corresponding permutation matrix. Then \(EA\) is obtained from \(A\) by the same swap: the \(i\)th row of \(EA\) equals the \(j\)th row of \(A\), the \(j\)th row of \(EA\) equals the \(i\)th row of \(A\), and all other rows are unchanged. Hence \(EA=B\).
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(2) Multiplying row \(i\) by a nonzero scalar \(c\). Here \(E\) is obtained from \(I_m\) by multiplying row \(i\) by \(c\), so \(E\) equals \(I_m\) except that \(E_{ii}=c\). Then the \(i\)th row of \(EA\) is \(c r_i\), and every other row of \(EA\) equals the corresponding row of \(A\). Thus \(EA=B\).
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(3) Replacing row \(i\) by row \(i\) plus \(c\) times row \(j\) (\(i\neq j\)). Here \(E\) is obtained from \(I_m\) by replacing row \(i\) with row \(i + c\) row \(j\), so \(E\) equals \(I_m\) except that \(E_{ij}=c\). Then the \(i\)th row of \(EA\) is \(r_i + c r_j\), and all other rows are unchanged. Hence \(EA=B\).
This proves the existence statement for row operations. The bracketed column statement is proved similarly by right-multiplication: if \(B\) is obtained from \(A\) by an elementary column operation and \(E\) is obtained from \(I_n\) by the same column operation, then \(AE\) has exactly the corresponding effect on the columns of \(A\), so \(B=AE\).
For the converse, let \(E\) be an elementary \(m\times m\) matrix. By definition, \(E\) is obtained from \(I_m\) by one elementary row operation, say \(\mathcal{R}\). But the argument above shows that for every \(A\), the product \(EA\) is exactly the result of applying \(\mathcal{R}\) to the rows of \(A\). Hence \(EA\) is obtained from \(A\) by the same elementary row operation that produces \(E\) from \(I_m\). The column case (with \(AE\)) is analogous.
Example 15.4 : Elementary operations and elementary matrices
Let
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(Type 1)
Interchanging the second row of \(A\) with the first row is an example of an elementary row operation of type 1. The resulting matrix is\[ B=\left(\begin{array}{rrrr} 2 & 1 & -1 & 3 \\ 1 & 2 & 3 & 4 \\ 4 & 0 & 1 & 2 \end{array}\right) \]Performing this same operation on \(I_{3}\), we obtain the elementary matrix
\[ E=\left(\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right) \]Note that \(E A=B\).
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(Type 2)
Multiplying the second column of \(A\) by \(3\) is an example of an elementary column operation of type 2. The resulting matrix is\[ C=\left(\begin{array}{rrrr} 1 & 6 & 3 & 4 \\ 2 & 3 & -1 & 3 \\ 4 & 0 & 1 & 2 \end{array}\right). \]Performing this same operation on \(I_{4}\), we obtain the elementary matrix
\[ E=\left(\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) \]Observe that \(A E=C\).
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(Type 3)
Adding \(4\) times the third row of \(A\) to the first row is an example of an elementary row operation of type 3. In this case, the resulting matrix is\[ M=\left(\begin{array}{rrrr} 17 & 2 & 7 & 12 \\ 2 & 1 & -1 & 3 \\ 4 & 0 & 1 & 2 \end{array}\right) \]Performing this same operation on \(I_{3}\), we obtain the elementary matrix
\[ E=\left(\begin{array}{lll} 1 & 0 & 4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right) \]Note that \(E A=M\).
Theorem 15.5 : Elementary matrices are invertible.
Elementary matrices are invertible, and the inverse of an elementary matrix is an elementary matrix of the same type.
Proof
Let \(E\) be an elementary \(n \times n\) matrix. Then \(E\) can be obtained by an elementary row operation on \(I_{n}\). By reversing the steps used to transform \(I_{n}\) into \(E\), we can transform \(E\) back into \(I_{n}\). The result is that \(I_{n}\) can be obtained from \(E\) by an elementary row operation of the same type. By Theorem 15.3, there is an elementary matrix \(\bar{E}\) such that \(\bar{E} E=I_{n}\). Therefore, by Exercise 11.10, \(E\) is invertible and \(E^{-1}=\bar{E}\).
Exercise¶
Exercise 15.8
Prove that if a matrix \(Q\) can be obtained from a matrix \(P\) by an elementary row operation, then \(P\) can be obtained from \(Q\) by an elementary row operation of the same type. Hint: Treat each type of elementary row operation separately.
Exercise 15.9
Prove that any elementary row [column] operation of type 1 can be obtained by a succession of three elementary row [column] operations of type 3 followed by one elementary row [column] operation of type 2.
Exercise 15.12
Let \(A\) be an \(m \times n\) matrix. Prove that there exists a sequence of elementary row operations of types 1 and 3 that transforms \(A\) into an upper triangular matrix.