2.8. Elementary Matrices

Definition - Elementary Matrices and Row Operations

An \(n\times n\) matrix \(E\) qualifies as an elementary matrix if it can be derived from the identity matrix \(I_n\) by performing a singular row operation. These operations encompass actions such as interchanging two rows, multiplying a row by a scalar factor, or adding one row to another. It’s worth noting that matrices generated exclusively through row exchanges of the identity matrix are specifically referred to as permutation matrices [Kuttler and Farah, 2020, Nicholson, 2018].

For example, the matrix

\[\begin{align*} E=\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \end{align*}\]

is the elementary matrix obtained from switching the 2nd and 3rd rows of \(I_{3}\). The matrix

\[\begin{align*} E=\begin{bmatrix} 3 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \end{align*}\]

is the elementary matrix obtained from multiplying the first row of \(I_{3}\) by 3. The matrix

\[\begin{align*} E=\begin{bmatrix} 1 & 0 \\ 5 & 1 \\ \end{bmatrix} \end{align*}\]

is the elementary matrix obtained from adding \(5\) times the first row to the second row of \(I_{2}\).

Multiplication by an Elementary Matrix and Row Operations

Consider an elementary matrix denoted as \(E\), which is derived from the identity matrix via a specific row operation. Let \(A\) represent a matrix. The product \(EA\) applies precisely the same row operations to the matrix \(A\) [Kuttler and Farah, 2020, Nicholson, 2018].

Action of Permutation Matrix

Let \(P^{ij}\) be a permutation matrix, which is essentially an elementary matrix responsible for exchanging the \(i^{\text{th}}\) and the \(j^{\text{th}}\) rows. When you multiply \(P^{ij}\) by matrix \(A\), the result is a matrix obtained from \(A\) by interchanging the \(i^{\text{th}}\) and the \(j^{\text{th}}\) rows [Kuttler and Farah, 2020, Nicholson, 2018].

Example: Let

\[\begin{align*} P^{23}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix}, ~A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}. \end{align*}\]

Find \(B\) where \(B = P^{23}A\)

Solution: \(B\) can be obtained from switching the last two rows of matrix \(A\). Therefore,

\[\begin{align*} B=\begin{bmatrix}1 & 2 \\ 5 & 6 \\ 3 & 4 \end{bmatrix} \end{align*}\]

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Multiplication by a Scalar and Elementary Matrices

Let \(E(k,i)\) represent the elementary matrix corresponding to a row operation in which the \(i^{\text{th}}\) row is multiplied by the nonzero scalar \(k\). In this scenario, the product \(E(k,i)A = B\), where matrix \(B\) is derived from matrix \(A\) by scaling the \(i^{\text{th}}\) row of \(A\) by a factor of \(k\) [Kuttler and Farah, 2020, Nicholson, 2018].

Action of Permutation Matrix

In the case of \(E(k\times \, i+ \, j)\), this elementary matrix is formed by adding \(k\) times the \(i^{\text{th}}\) row to the \(j^{\text{th}}\) row of the identity matrix \(I\). Consequently, when you multiply \(E(k\times \, i+ \, j)\) by matrix \(A\), the result is denoted as \(B\), and it is obtained from matrix \(A\) by adding \(k\) times the \(i^{\text{th}}\) row to the \(j^{\text{th}}\) row of \(A\) [Kuttler and Farah, 2020, Nicholson, 2018].

Example: Let

\[\begin{align*} A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}. \end{align*}\]

Find \(B = E(3,1)A\) and \(C = E(4\times \, 2+ \, 3)A\)

Solution: \(B\) can be obtained from multiplying the first row of matrix \(A\) by 3 and replacing it with the first row of \(A\). Therefore,

\[\begin{align*}B=\begin{bmatrix}3 & 6 \\ 3 & 4 \\ 5 & 6\end{bmatrix}.\end{align*}\]

Moreover, \(C\) can be obtained from multiplying the second row of matrix \(A\) by 4 and adding it to the last row of \(A\). Therefore,

\[\begin{align*}C=\begin{bmatrix}1 & 2 \\ 3 & 4 \\ 17 & 22\end{bmatrix}.\end{align*}\]

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Theorem: Elementary Matrices and Inverses

The inverses of elementary matrices do exist, and it’s noteworthy that these inverses are themselves elementary matrices [Kuttler and Farah, 2020, Nicholson, 2018].