Glen A. Hagemann and Joseph R. Cummins
Matrix Addition
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1. In order to add matrices the matrices must have the same dimensions.
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2. When adding matrices we add corresponding entries from the matrices. The sum of these entries listed in a new matrix is the sum of the matrices.
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3. The sum of any matrices is a matrix with the same dimension as the dimensions of the matrices being added.
Examples:
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Solution:
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Solution:
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In sample #2, the sum is a zero matrix because the matrices being added are opposites (additive inverses) of each other, and when opposites are added zero is always the sum.
Sample #3
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Add:
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Solution: These matrices cannot be added because they do not have the sam e dimensions.
Matrix subtraction
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1. As in addition, in order to find the difference between two matrices, the matrices must have the same dimensions.
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2. In order to find the difference between two matrices we find the difference between corresponding elements.
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3. Change each entry in the matrix which is the subtrahend to its additive inverse (opposite) and then change the sign of operation to addition and then simply follow the rules for adding matrices.
Examples:
Sample #1 Simplify:
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Sample #2 Simplify:
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Sample #3 Simplify:
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This example cannot be simplified because the matrices do not have the same dimensions.
Matrix multiplication
There are two kinds of products that involve matrices The first is a scaler product which is the product of a real number ( a scaler ) and a matrix. The product is a matrix in which the entry in any particular address is simply the product of the scaler and the entry in the same address in the given matrix.
Examples of a scaler product:
Sample #1
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Sample #2
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The other matrix multiplication product is the product of two matrices.
Multiplication of a row matrix by a-column matrix
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1. The number of entries in the row matrix must equal the number of entries in the column matrix or it is not possible to multiply.
Examples:
Sample #1
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Sample #2
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Sample #3
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Sample #3 is an example where multiplication cannot take place because the matrices do not have the same number of entries.
Multiplication of a matrix by a column matrix
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1. The result of this type of multiplication is always a column matrix with the same number of entries as the number of rows in the non-column matrix. The product’s first entry is obtained by multiplying R1 of the first matrix by the column matrix. The second entry of the product is obtained by multiplying R2 of the first matrix by the column matrix. This progression is continued until the final entries are multiplied.
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2. In order to do this multiplication, the number of columns must equal the number of entries in the column matrix.
Examples:
Sample #1
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The product will be a column matrix. We compute the product as follows.
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Therefore, the first entry in the column matrix (product) is 12. We then continue to compute the product.
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Thus the answer is:
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Sample #2
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Multiply:
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Since the first matrix has only 3 columns and the column matrix has 4 entries these matrices cannot be multiplied.
Multiplication of one matrix by another
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1. First of all, the number of columns in the first matrix must equal the number of rows in the second matrix or the matrices cannot be multiplied.
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2. If the requirement above is met, then multiply the first matrix and the first column of the second matrix. List the resulting column matrix. If there are other unused columns in the second matrix, multiply the first matrix and the next column of the second matrix. List the resulting column matrix. Continue until completed, then form a matrix whose columns are the listed columns, written in the order obtained. This matrix is the answer.
Examples:
Sample #1
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Simplify:
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Sample #2
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Simplify:
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Sample #3
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Since the first matrix has 4 columns an d the second matrix has 3 rows, they cannot be multiplied.
Solving linear systems
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1. When you have the system of two linear equations
where A= the system of two linear
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equations may be represented by the matrix equation AX=C.
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2. When A is invertible, this matrix equation has an unique solution; X= A
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C.
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3. Therefore, when A is invertible the linear system has a unique solution. The solution may be found by solving the matrix equation.
Example:
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Solve:
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4x Ðy = 5
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3x + 2y = 12
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The above system may be represented by the matrix equation:
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Since has an inverse. Therefore, the
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equation has a unique solution.
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Therefore, x=2 and y=3
