Covers the rules and process for adding and subtracting matrices. This stack emphasizes the condition that matrices must have identical dimensions and explains the element-wise nature of these operations.
What is the fundamental condition required to add or subtract two matrices?
The two matrices must have the exact same dimensions. That is, they must have the same number of rows and the same number of columns.
How do you add two matrices, \(A\) and \(B\), to find their sum \(C\)?
You add the corresponding elements from each matrix. If \(C = A + B\), then each element of \(C\) is given by \(c_{ij} = a_{ij} + b_{ij}\).
How do you subtract matrix \(B\) from matrix \(A\) to find their difference \(D\)?
You subtract the elements of \(B\) from the corresponding elements of \(A\). If \(D = A - B\), then each element of \(D\) is given by \(d_{ij} = a_{ij} - b_{ij}\).
Can you add a \(2 \times 3\) matrix and a \(3 \times 2\) matrix? Why or why not?
No. The matrices do not have the same dimensions, so addition (and subtraction) is undefined. The number of rows and columns must match exactly.
What is the zero matrix, often denoted by \(O\)?
The zero matrix is a matrix of any dimension where every single element is \(0\). It acts as the additive identity in matrix addition.
What is the additive identity property for matrix addition?
For any matrix \(A\), adding the zero matrix \(O\) of the same dimensions results in the original matrix \(A\). The property is written as \(A + O = A\).
What is the additive inverse of a matrix \(A\)?
The additive inverse of \(A\) is the matrix \(-A\), where each element is the negative of the corresponding element in \(A\). Their sum is the zero matrix: \(A + (-A) = O\).
Does matrix addition follow the commutative property? In other words, is \(A + B = B + A\)?
Yes. Since addition is performed element-wise and real number addition is commutative (\(a_{ij} + b_{ij} = b_{ij} + a_{ij}\)), matrix addition is also commutative.
Does matrix addition follow the associative property? In other words, is \((A + B) + C = A + (B + C)\)?
Yes. Just like with the commutative property, this holds because real number addition is associative. You can group the matrices differently without changing the final sum.
