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Matrices

Matrix Multiplication

Problems

Problems

Scalar Multiplication

To multiply a matrix by a scalar, that is, a single constant, variable, or expression, multiply all the entries in the matrix by the scalar:

± =    


For example,

5 =    

Scalar multiplication is distributive: ±(A + B) = ±A + ±B . For example,

4( + ) = 4 +4    

Multiplication of Two Matrices

To multiply two matrices, we first must know how to multiply a row (a p matrix) by a column (a p×1 matrix). To multiply a row by a column, multiply the first entry of the row by the first entry of the column. Then multiply the second entry of the row by the second entry of the column, and so on, and add all the results. The answer should be a single number. For example,

[ 8 –1 3 0 ] = 8(2) + (- 1)(6) + 3(- 4) + 0(- 2) = 16 - 6 - 12 + 0 = - 2    

A row can be multiplied by a column if and only if the row and the column have the same number of entries. Similarly, two matrices can be multiplied if and only if the first matrix has the same number of columns as the second matrix has rows. That is, two matrices can be multiplied if and only if they have the dimensions m×p and p×n . The answer will be a matrix with the same number of rows as the first matrix and the same number of columns as the second matrix. In other words, it will be of dimension m×n .

PARGRAPH To multiply two matrices, multiply each row in the first matrix by each column in the second matrix. Place the result of multiplying the 1st row by the 1st column in the 1st row and 1st column of the answer matrix. Place the result of multiplying the 1st row by the 2nd column in the 1st row and second column of the answer matrix. In general, place the result of multiplying row i by column j in row i and column j of the answer matrix.

Here is an example of matrix multiplication:

= ?    


Row 1, column 1:

= 1(5) + 6(2) + - 2(- 1) = 5 + 12 + 2 = 19    



Row 1, column 2:

= 1(0) + 6(4) + (- 2)(- 2) = 0 + 24 + 4 = 28    



Row 1, column 3:

= 1(- 1) + 6(0) + (- 2)(4) = - 1 + 0 - 8 = - 9    



Row 1, column 4:

= 1(1) + 6(6) + (- 2)() = 1 + 36 - 1 = 36    



Row 2, column 1:

= 0(5) + (- 3)(2) + 10(- 1) = 0 - 6 - 10 = - 16    



and so on...

Thus, the answer is

.    

Note that a 2×3 matrix times a 4 matrix is a 2×4 matrix.

Matrix multiplication is not necessarily commutative: it is not always true that AB = BA . Matrix multiplication is, however, associative: (AB)C = A(BC) .

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