unit 3: matrices. matrix: a rectangular arrangement of data into rows and columns, identified by...
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Unit 3: Matrices
Matrix: A rectangular arrangement of data into rows and columns, identified by capital letters.
Matrix Dimensions: Number of rows, m, by the number of columns, n. Read as “m by n” matrix. Also known as the order of a matrix.
◦RBC (ROWS BY COLUMNS)
Determine the dimensions of each matrix.
Elements
Matrix Element: Each number in a matrix, identified by its row and column.
Example: amn
Refers to the m-th row and n-th column
Example
Identify each element.
1. a23
2. a12
3. a31
4. a21
Adding and Subtracting Matrices
When matrices have the same dimension you add and subtract them by adding or subtracting each corresponding element.
Add or Subtract the following matrices:
Scalar Multiplication
Matrix MultiplicationWhen multiplying matrices two
matrices find the dimensions of each:
Is it possible to multiply these matrices? If so, what would the dimension of your answer matrix be?
1)
2)
Multiplying Matrices
Can the following Matrices be multiplied? If so, what dimensions will the product be??
1. x
2. x
How to multiply matrices
Multiply the elements of each row in the first matrix by the elements in each column of the second matrix
Add the products to get the new element.
Matrix Multiplication
Equivalent Matrices
DETERMINANT OF MATRICES
A special number that can be calculated from the matrix.
It tells us things about the matrix that are useful in systems of linear equations, in calculus, and more
The symbol for determinant is two vertical lines either side
Determinant of a Matrix
Determinant of a 2x2
Find the determinant of the following 2x2 matrices:
Determinant of a 3x3 Matrix
Find the determinant of the following.
MATRIX EQUATIONS
Matrix Equation Example
Solve each equation:
INVERSE OF MATRICES
For matrices, there is no such thing as division. You can add, subtract, and multiple matrices, but you cannot divide them.
There is a related concept called inversion
AX=C
Using Inverses to Solve For X
Inverse Notation
REMEMBER we denote inverse with a -1 power
So the inverse of matrix A is A-1
Requirement to have an InverseMatrix MUST be square, meaning it has the same number of rows and columns
Matrix MUST NOT have a determinant of zero.
Inverse exist?!
Does the inverse exist?!?!
Multiplying InverseWhen you Multiply a matrix A times it’s inverse, the Product is the Identity Matrix.
Identity Matrix is a square matrix where the top left to Bottom right diagonal are all ones, and everything else is a zero
Determine if the following matrices are inverses. 1.
2.
Finding the Inverse of a 2x2
IF
THEN
Find the inverse of the following matrix.
Use your calculator!
1. 2nd Matrix Edit2. Put in your matrix3. 2nd Matrix NAME4. Get your matrix5. X-1
The inverse of a matrix can be used when solving matrix equations.
For Matrices A and B, we can find Matrix X:
IF AX = B
THEN X = A-1B
*Solve for X:
X = A-
1B
You Try! Solve Each Matrix Equation:
Solutions: