FOCUS: Students will be encouraged to develop an understanding of matrices and an appreciation of the diverse nature of matrix algebra and its value in solving problems from all branches of mathematics.
MX 1. Definition of a Matrix
1.1 Know uses of a matrix as a data storage mechanism and as a mathematical tool.
1.2 Understand matrix terminology; element, row, column, index, dimension, square matrix, diagonal matrix, triangular matrix.
1.3 Identify and use vector matrices.
MX 2. Basic Operations on Matrices
2.1 Know how to add and subtract matrices and understand that only matrices with the same dimensions can be added and subtracted.
2.2 Understand the properties of matrix addition, the definition of the additive identity (zero matrix O_{m,n} ), the additive inverse (-A), commutative and associative properties.

Understand the properties of scalar multiplication and the fact that the commutative and associative laws work for this type of multiplication as well as the distributive law for scalar multiplication over matrix addition.

2.4 Understand that the transpose of an m \times n   matrix is an n \times m   matrix.
2.5 Simplify and solve matrix expressions involving addition, subtraction, scalar multiplication and transposes of matrices.
MX 3. Multiplication of Matrices
3.1 Understand that even though matrix multiplication appears strange, it is useful at even low levels of application.
3.2 Know that matrix multiplication is associative, but not commutative, even on square matrices.
3.3 Know the definition of the identity matrix I_{m,n} and that I A = A I = A .
MX 4. Determinants
4.1 Understand not only how to calculate the determinant of a matrix, but how the properties of determinants, particularly with respect to linear dependence affects determinants.
4.2 Know how to calculate the determinant of a matrix with arrows.
4.3 Know and understand that calculating determinants of a matrix with cofactors is both efficient and extendable to higher order matrices, particularly if they use arrows.
4.4 Know the definition of and identify singular and non-singular matrices.
MX 5. Inverse Matrices
5.1 Know the definition of the inverse matrix and its properties.
5.2 Know the definition of the adjoint matrix and use one or more methods to find the inverse of a non-singular matrix. For example, inv({\bf{A}}) = \frac{1}{{\det ({\bf{A}})}} \cdot \left( {adj({\bf{A}})^T } \right) .
MX 6. Linear Systems and Matrices
6.1 Can state and use Cramer’s Rule for using determinants to solve linear 2 \times 2 systems and linear 3 \times 3 systems and understand how to check and/or interpret their results graphically.
6.2 Can relate properties of determinants to linear systems that are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions.
6.3 Can reduce a matrix to row echelon form using Jordanian Elimination.
6.4 Can use Gauss-Jordan Elimination for solving linear systems and linear systems and understand how to check and/or interpret their results graphically.  They also understand how this method relates to solving systems using a combination of the addition and substitution methods.
6.5 Solve real-world problems involving systems of linear equations with matrices.
MX 7. Elements of Vector Algebra
7.1 Understand the definition of vectors, zero vectors, unit vectors, radius vectors (position vectors), displacement vectors and the definitions of

\vec i,\,\,\vec j,\,\,\vec k .


Demonstrate an understanding of the geometric interpretation of vectors, vector addition (by means of parallelograms) and scalar multiplication in the plane and in three-dimensional space.

7.3 Understand the idea of linear dependence and independence of vectors and that two (three) vectors are collinear (coplanar) if they are linearly dependent.

Are familiar with the geometric interpretations of the determinants of 2 \times 2 and 3 \times 3 matrices as the area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dimensional spaces.

7.5 Compute the scalar (dot) product of two vectors in n-dimensional space and know that perpendicular vectors have zero dot product.
7.6 Solve real-world problems using vector and matrix notation and algebra.
Framework & Comments
Standard MX1.0: In order to give the student an understanding of where matrices come from, the definition of a matrix should be along the lines of the following sample SOS Math.

Matrices, though they may appear weird objects at first, are a very important tool in expressing and discussing problems which arise from real life cases.
Our first example deals with economics. Indeed, consider two families A and B (though we may easily take more than two). Every month, the two families have expenses such as: utilities, health, entertainment, food, etc... Let us restrict ourselves to: food, utilities, and health. How would one represent the data collected? Many ways are available but one of them has an advantage of combining the data so that it is easy to manipulate them. Indeed, we will write the data as follows:

Month = \left( {\matrix{ {Family} & {Food} & {Utilities} & {Health} \cr A & a & b & c \cr B & d & e & f \cr } } \right)

If we have no problem confusing the names and what the expenses are, then we may write

M = \left( {\matrix{ a & b & c \cr d & e & f \cr } } \right)

This is what we call a Matrix.”

Standard MX3.0: Matrix multiplication should be introduced and motivated via a sample as below SOS Math.

Before we give the formal definition of how to multiply two matrices, we will discuss an example from a real life situation. Consider a city with two kinds of population: the inner city population and the suburb population. We assume that every year 40% of the inner city population moves to the suburbs, while 30% of the suburb population moves to the inner part of the city.

Let I_o be the initial population of the inner city and S_o be the initial population of the suburban area.
So after one year, the population of the inner part is I_1=0.6 I_o+0.3S_o , while the population of the suburbs is S_1=0.4I_o+0.7S_o .
After two years, the population of the inner city is I_2 =0.6 (0.6 I_o + 0.3 S_o) + 0.3 (0.4 I_o + 0.7 S_o) and the suburban population is given by S_2 =0.4 (0.6 I_o + 0.3 S_o) + 0.7(0.4 I_o + 0.7 S_o) .
Is there a nice way of representing the two populations after a certain number of years? Let us show how matrices may be helpful to answer this question. Let us put the coefficients in one matrix and the two populations in a table (meaning a column object with two entries):
the coefficients: \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) and the populations: \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) .
Then, \left( {\matrix{ {I_1 } \cr {S_1 } \cr } } \right) = \left( {\matrix{ {0.6I_o + 0.3S_o } \cr {0.4I_o + 0.7S_o } \cr } } \right) .

If we consider the following rule for matrix multiplication: \left( {\matrix{ a & b \cr c & d \cr } } \right) \cdot \left( {\matrix{ I \cr S \cr } } \right) = \left( {\matrix{ {aI + bS} \cr {cI + dS} \cr } } \right) ,

we have: \left( {\matrix{ {I_1 } \cr {S_1 } \cr } } \right) = \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right)   and   \left( {\matrix{ {I_2 } \cr {S_2 } \cr } } \right) = \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) .

Standard MX5.0 Motivation for learning how to find inverse matrices and indeed for learning how to do row operations can come from a sample such as the following  SOS Math.

There are many ways to encrypt a message. And the use of coding has become particularly significant in recent years (due to the explosion of the internet for example). One way to encrypt or code a message uses matrices and their inverse. Indeed, consider a fixed invertible matrix A. Convert the message into a matrix B such that AB is possible to perform. Send the message generated by AB. At the other end, they will need to know A^{-1} in order to decrypt or decode the message sent. Indeed, we have: A^{-1}(AB)=B , which is the original message.

Keep in mind that whenever an undesired intruder finds A, we must be able to change it. So we should have a mechanical way of generating simple matrices A which are invertible and have simple inverse matrices. Note that, in general, the inverse of a matrix involves fractions which are not easy to send in an electronic form. The best is to have both A and its inverse with integers as their entries. In fact, we can use our previous knowledge to generate such class of matrices. Indeed, if A is a matrix such that its determinant is +1 or -1 and all its entries are integers, then A-1 will have entries which are integers.

So how do we generate such class of matrices? One practical way is to start with an upper triangular matrix with +1 or -1 on the diagonal and integer-entries. Then we use the elementary row operations to change the matrix while keeping the determinant unchanged. Do not multiply rows with non-integers while doing elementary row operations. Let us illustrate this on an example.

Example. Consider the matrix \left( {\matrix{ { - 1} & 5 & { - 1} \cr 0 & 1 & 8 \cr 0 & 0 & 1 \cr } } \right) .

First we keep the first row and add it to the second as well as to the third rows. We obtain \left( {\matrix{ { - 1} & 5 & { - 1} \cr { - 1} & 6 & 7 \cr { - 1} & 5 & 0 \cr } } \right) .

Next we keep the first row again, we add the second to the third, and finally add the last one to the first multiplied by -2. We obtain \left( {\matrix{ { - 1} & 5 & { - 1} \cr { - 2} & {11} & 7 \cr 1 & { - 5} & 2 \cr } } \right)

This is our matrix A. Easy calculations will give det(A) = -1, which we knew since the above elementary operations did not change the determinant from the original triangular matrix which obviously has -1 as its determinant. We leave the details of the calculations to the reader. The inverse of A is \left( {\matrix{ {57} & { - 5} & {46} \cr {11} & { - 1} & 9 \cr { - 1} & 0 & { - 1} \cr } } \right)

Back to our original problem. Consider the message “I LOVE MONICA”

To every letter we will associate a number. The easiest way to do that is to associate 0 to a blank or space, 1 to A, 2 to B, etc...

Another way is to associate 0 to a blank or space, 1 to A, -1 to B, 2 to C, -2 to D, etc...

Let us use the second choice. So our message is given by the string

I       L   O   V   E       M   O   N   I   C   A
5  0  -6   8  -11  3   0   7   8   -7   5   2   1

Now we rearrange these numbers into a matrix B (with 3 rows so that we can multiply AB). For example, we have B = \left( {\matrix{ 5 & 8 & 0 & { - 7} & 1 \cr 0 & { - 11} & 7 & 5 & 0 \cr { - 6} & 3 & 8 & 2 & 0 \cr } } \right)

Then we perform the product AB, where A is the matrix found above. We get

AB = \left( {\matrix{ { - 1} & 5 & { - 1} \cr { - 2} & {11} & 7 \cr 1 & { - 5} & 2 \cr } } \right)\left( {\matrix{ 5 & 8 & 0 & { - 7} & 1 \cr 0 & { - 11} & 7 & 5 & 0 \cr { - 6} & 3 & 8 & 2 & 0 \cr } } \right) = \left( {\matrix{ 1 & { - 66} & {27} & {30} & { - 1} \cr { - 52} & { - 116} & {133} & {83} & { - 2} \cr { - 7} & {69} & { - 19} & { - 28} & 1 \cr } } \right)

The encrypted message to be sent is: 1, -52, -7, -66, …, -1, -2, 1

Mapping with Other Standards for Matrix/Linear Algebra
Intl Standards for Matrix Algebra CA - California Standards for Linear Algebra
MX 6.4 1.0 Students solve linear equations in any number of variables by using Gauss-Jordan elimination.
MX 6.4 2.0 Students interpret linear systems as coefficient matrices and the Gauss-Jordan method as row operations on the coefficient matrix.
MX 6.3 3.0 Students reduce rectangular matrices to row echelon form.
MX 2.1, MX 2.2, MX 2.3, MX 2.5 4.0 Students perform addition on matrices and vectors.
MX 2.4, MX 3.1, MX 3.2, MX 3.3 5.0 Students perform matrix multiplication and multiply vectors by matrices and by scalars.
ALG 1 3.3, MX 6.2 6.0 Students demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions.
MX 7.2 7.0 Students demonstrate an understanding of the geometric interpretation of vectors and vector addition (by means of parallelograms) in the plane and in three-dimensional space.
ALG 1 3.1 8.0 Students interpret geometrically the solution sets of systems of equations. For example, the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two-by-two system is interpreted as the intersection of a pair of lines in the plane.
MX 5.1, MX 5.2 9.0 Students demonstrate an understanding of the notion of the inverse to a square matrix and apply that concept to solve systems of linear equations.
MX 4.1, MX 4.2, MX 4.3, MX 4.4

MX 7.4

10.0 Students compute the determinants of 2 x 2 and 3 x 3 matrices and are familiar with their geometric interpretations as the area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dimensional spaces.
MX 5.2 11.0 Students know that a square matrix is invertible if, and only if, its determinant is nonzero. They can compute the inverse to 2 x 2 and 3 x 3 matrices using row reduction methods or Cramer's rule.
MX 7.5 12.0 Students compute the scalar (dot) product of two vectors in n- dimensional space and know that perpendicular vectors have zero dot product.

Personal comment by LFS on the California Linear Algebra Standards: In my opinion, the CA standards for Linear Algebra only appear to cover less material than the International Standard for Matrix Algebra. In fact, the opposite is true. Each CA standard - by its very conciseness - fails to fully expose the underlying concepts and content that need to be understood in order to accomplish its goal. My personal goal in the International Standard was to do exactly this.

This set of standards is still under development. Please send any suggestions, comments and critisms to contact.

Developed by: RGM, LFS, and TRF.

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Page last modified on September 16, 2008, at 01:52 PM