Operations with Matrices: Introduction to Determinant, Solving Systems, THE INVERSE OF A SQUARE MATRIX, THE MATRIX EQUATION
6.2 - Operations with Matrices
Two matrices are equal if and only if
The order of the matrices are the same
The corresponding elements of the matrices are the same
AdditionOrder of the matrices must be the same
Add corresponding elements together
Matrix addition is commutative
Matrix addition is associativ
SubtractionThe order of the matrices must be the same
Subtract corresponding elements
Matrix subtraction is not commutative (neither is subtraction of real numbers)
Matrix subtraction is not associative (neither is subtraction of real numbers)
A scalar is a number, not a matrix.
The matrix can be any order
Multiply all elements in the matrix by the scalar
Scalar multiplication is commutative
Scalar multiplication is associative
Matrix of any order
Consists of all zeros
Denoted by capital O
Additive Identity for matrices
Any matrix plus the zero matrix is the original matrix
Am×n × Bn×p = Cm×p
Matrix multiplication is not commutative
There is no matrix division.
Identity matrix of size 3
Properties of Matrices
Properties of Real Numbers that aren't Properties of Matrices
Commutativity of Multiplication
You can not change the order of a multiplication problem and expect to get the same product. AB≠BA
You must be careful when factoring common factors to make sure they are on the same side. AX+BX = (A+B)X and XA + XB = X(A+B) but AX + XB doesn't factor.
Evaluating a Function using a Matrix
Consider the function f(x) = x2 - 4x + 3 and the matrix A
The initial attempt to evaluate the f(A) would be to replace every x with an A to get f(A) = A2 - 4A + 3. There is one slight problem, however. The constant 3 is not a matrix, and you can't add matrices and scalars together. So, we multiply the constant by the Identity matrix.
f(A) = A2 - 4A + 3I.
Evaluate each term in the function and then add them together.
Factoring Expressions
Some examples of factoring are shown. Simplify and solve like normal, but remember that matrix multiplication is not commutative and there is no matrix division.
2X + 3X = 5X
AX + BX = (A+B)X
XA + XB = X(A+B)
AX + 5X = (A+5I)X
AX+XB does not factor
A system of linear equations can be written as AX=B where A is the coefficient matrix, X is a column vector containing the variables, and B is the right hand side. We'll learn how to solve this equation in the next section.
If there are more than one system of linear equations with the same coefficient matrix, then you can expand the B matrix to have more than one column. Put each right hand side into its own column.