Algebra II End-of-Course Exam Content Standards—Module: Matrices
Successful students will be able to compute with and use matrices to organize information, solve systems of equations, and perform transformations of geometric figures. They will use and interpret matrix notation to represent a vector and perform operations on vectors and matrices. There are a variety of types of test items including some that cut across the objectives in this standard and require students to make connections and solve contextual problems.
M1. Matrix arithmetic
a. Perform addition, subtraction, and scalar multiplication of matrices.
- Recognize that matrix addition is associative and commutative.
- Know and fluently use matrix notation for rows, columns, and entries of cells.
b. Perform matrix multiplication.
- Describe the characteristics of matrices that can be multiplied and those that cannot.
- Recognize that matrix multiplication is not commutative and provide examples of square matrices A and B such that AB ≠ BA.
- Recognize that matrix multiplication is distributive with respect to addition.
- Apply the associative property of matrix multiplication
The associative property of matrix multiplication states that if there are three matrices, A, B, and C such that AB and BC are defined, then (AB)C and A(BC) are defined and (AB)C=A(BC).
- Know and use the algorithm for matrix multiplication.
M2. Solving systems of equations using matrices
a. Find the determinant of a 2x2 or 3x3 matrix.
b. Determine the inverse of a 2x2 or 3x3 matrix or indicate that no inverse exists.
- Know and use the definition and properties of the identity matrix.
- Find the inverse of a 2x2 matrix if the inverse exists.
- Use row reduction to find the inverse of a 3x3 matrix when it exists.
c. Represent 2-variable and 3-variable systems of linear equations using matrices and use them to solve the system.
- Translate between equation and matrix notation of a system of linear equations.
- Use both technology and by-hand methods, including row reduction, substitution, and elimination, to solve systems involving 3 equations.
- 3x3 systems to be solved by hand should have integer or simple
fractional solutions and small integral coefficients.
d. Solve a matrix equation.
- Use the inverse of a matrix to solve a matrix equation.
M3. Matrix transformations
a. Use matrix tools to represent and transform geometric objects in the coordinate plane.
- Use 2x2 matrices to represent transformations in the coordinate
Example: represents the rotation of the point A(3, 1) 180° about the origin.
Example: represents the rotation of the point A(3, 1) over the y-axis.
a. Represent vectors as matrices in two dimensions.
- Use either horizontal or vertical representation of vectors, with an understanding of both.
- Use vectors to represent and transform geometric objects in the coordinate plane.
- Know the relationship between the ordered pair representation of a vector and its graphical representation.
b. Add, subtract, and compute the dot product of twodimensional vectors; multiply a two-dimensional vector by a scalar.
- Represent addition, subtraction, and scalar multiplication of vectors graphically.
- Use the dot product of two vectors to determine whether they are
normal (perpendicular) to each other.