Summarizes the key algebraic properties of matrix operations, including associativity and distributivity. This stack highlights crucial differences from real number algebra, such as the non-commutativity of multiplication and the role of the identity matrix.
Is matrix multiplication commutative? In other words, does \(AB = BA\) for any two matrices \(A\) and \(B\)?
No. In general, matrix multiplication is not commutative. \(AB\) is usually not equal to \(BA\). In many cases, one product may be defined while the other is not.
Is matrix multiplication associative? That is, for compatible matrices, is \((AB)C = A(BC)\)?
Yes. Matrix multiplication is associative. You can group the multiplication in any order, provided the dimensions are compatible for the operations to be defined.
What is the multiplicative identity matrix, denoted by \(I\)?
The identity matrix \(I\) is a square matrix with \(1\)s on the main diagonal (from top-left to bottom-right) and \(0\)s everywhere else. It acts like the number \(1\) in multiplication.
What is the property of the identity matrix \(I\) in matrix multiplication?
For any compatible matrix \(A\), multiplying by the identity matrix leaves \(A\) unchanged: \(AI = A\) and \(IA = A\).
Write down the \(2 \times 2\) identity matrix, \(I_2\).
State the left and right distributive properties for matrix multiplication over addition.
Left Distributive Property: \(A(B+C) = AB + AC\). Right Distributive Property: \((A+B)C = AC + BC\). Both forms are necessary because matrix multiplication is not commutative.
If \(AB = O\) (the zero matrix), does this imply that either \(A = O\) or \(B = O\)?
No. Unlike with real numbers, it is possible for the product of two non-zero matrices to be the zero matrix. This is a critical difference from standard algebra.
Given \(A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}\), what is \(AB\)?
$$ AB = \begin{pmatrix} 1-1 & -1+1 \\ 1-1 & -1+1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = O $$. This shows that two non-zero matrices can multiply to the zero matrix.
How do you calculate \(A^2\) for a matrix \(A\)? What condition must be met?
\(A^2\) is shorthand for \(A \times A\). For this to be defined, the number of columns of \(A\) must equal the number of rows of \(A\), meaning \(A\) must be a square matrix.
What is the transpose of a matrix, denoted \(A^T\)?
The transpose of a matrix \(A\) is formed by swapping its rows and columns. The element at row \(i\), column \(j\) of \(A\) becomes the element at row \(j\), column \(i\) of \(A^T\).
If \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix}\), what is \(A^T\)?
