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An \(m\times n\) matrix A is a rectangular array with \(m\) and \(m\) columns. We denote the entry in the i-th row and j-th column by \(a_{ij}\). That is, \[\left[\begin{array}{cccccc} a_{11} & a_{12} & \ldots & a_{1j} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2j} & \dots & a_{2n} \\ \vdots & \vdots && \vdots && \vdots \\ a_{i1} & a_{i2} & \ldots & a_{ij} & \ldots & a_{in} \\ \vdots & \vdots && \vdots && \vdots \\ a_{m1} & a_{m2} & \ldots & a_{mj} & \ldots & a_{mn} \end{array}\right]\] Two \(m\times n\) matrices \(A\) and \(B\) are equal if \(a_{ij} =b_{ij}\) for all \(1\leq i\leq m\), \(1 \leq j\leq n\). The set of all \(m\times n\) matrices with real entries is denoted by \(M_{m\times n} (\mathbb{R})\). |
| A matrix in \(M_{n\times n}(\mathbb{R})\) (the number of rows of the matrix is equal to the number of columns) is called a square matrix. |
The following is a \(3\times 3\) square matrix.
\begin{bmatrix} a &b&c\\d&e&f\\g&h&i\end{bmatrix}
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A matrix \(U \in M_{n\times n}(\mathbb{R})\) is said to be upper triangular if \(u_{ij}=0\) whenever \(i>j\). A matrix \(L \in M_{n\times n}(\mathbb{R})\) is said to be lower triangular if \(l_{ij}=0\) whenever \(i<j\). |
Upper triangle matrices
\[\begin{bmatrix} 3&4\\0&1\end{bmatrix} \qquad \begin{bmatrix} 5&2&1\\0&2&3\\0&0&1\end{bmatrix} \]
Lower triangle matrices
\[\begin{bmatrix} -3&0\\4&1\end{bmatrix} \qquad \begin{bmatrix} 0&0&0\\1&2&0\\0&5&0\end{bmatrix} \]
Upper and Lower triangle matrix
\[ \begin{bmatrix} 3&0&0\\0&2&0\\0&0&1\end{bmatrix} \]
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Let \(A, B\in M_{n\times n} (mathbb{R})\). We define addition of matrices by \[(A+B)_{ij} = (A)_{ij} + (B)_{ij} \] We define scalar multiplication of \(A\) by a scalar \(t\in\mathbb{R}\) by \[(tA)_{ij}=t(A)_{ij}\] |
By the definition above, matrix addition can only be performed on matrices of the same size. See an example in the video.
Example: Multiply matrix \(A\) by the scalar multiple \(k=-3\)
\[A=\begin{bmatrix} 3 & 0 &2\\-2&0&0\\0&-3&4\end{bmatrix}\]
\[3A=\begin{bmatrix} -9 & 0 &-6\\6&0&0\\0&9&-12\end{bmatrix}\]
Columns and rows can be switched by finding the transpose of a matrix.
| Let \(A\in = M_{m\times n}\in(\mathbb{R})\). The transpose of \(A\), is the \(n\times m\) matrix, denoted \(A^T\), whose ij-th entry is the ji-th entry of \(A\). That is \[(A^T)_{ij} = (A)_{ji}\] |
Example: Find the transpose of \(A\)
\[A=\begin{bmatrix} -3&6&2\\0&-2&1\end{bmatrix}\]
Solution
\[A^T=\begin{bmatrix} -3&0\\6&-2\\2&1\end{bmatrix}\]
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Properties of Transpose Matrices If \(A\) and \(B\) are matrices, and \(s\in\mathbb{R}\), then
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Let \(A\in M_{m\times n}(\mathbb{R})\) with rows \(\vec{a}_1,\ldots,\vec{a}_m\) and let \(B\in M_{n\times p}(\mathbb{R})\) with columns \(\vec{b}_1,\ldots,\vec{b}_p\) . We define \(AB\) to be the \(m\times p\) matrix whose ij-th entry is \[(AB)_{ij} = \vec{a}_i \cdot \vec{b}_{j}\] |
There are key elements to notice during matrix multiplication:
When performing an operation with an identity, the resultant is the same value. For example, adding 0 to any number gives you the same number. Multiplying any number by 1 also gives you the same number. How do these identities look like with matrices? The addition identity is a matrix with 0 in all its elements. However, it is not true that having a matrix with all 1s as its elements will create a multiplication identity. Instead, the identity matrix for multiplication has 1s only in the diagonal and 0s everywhere else.
| The identity matrix for an \(n\times n\) matrix is \[I=\begin{bmatrix}1&\ldots&\ldots&0\\0&1&\ldots&0\\ \vdots&&\ddots&\vdots\\0&\ldots&\ldots&1\end{bmatrix}\] |
For example, the identity matrix \(I\) in a \(3\times 3\) matrix is
\begin{bmatrix} 1&0&0\\0&1&0\\0&0&1 \end{bmatrix}
The identity matrix \(I\) in a \(4\times 4\) matrix is
\begin{bmatrix} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1 \end{bmatrix}
