Lesson MATRIX MULTIPLICATION
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This lesson provides an overview of matrix multiplication. REFERENCES <a href = "http://people.hofstra.edu/Stefan_Waner/realWorld/tutorialsf1/frames3_2.html" target = "_blank">http://people.hofstra.edu/Stefan_Waner/realWorld/tutorialsf1/frames3_2.html</a> <a href = "http://www.purplemath.com/modules/mtrxmult.htm" target = "_blank">http://www.purplemath.com/modules/mtrxmult.htm</a> <a href = "http://www.mathwarehouse.com/algebra/matrix/multiply-matrix.php" target = "_blank">http://www.mathwarehouse.com/algebra/matrix/multiply-matrix.php</a> <a href = "http://www.intmath.com/Matrices-determinants/4_Multiplying-matrices.php" target = "_blank">http://www.intmath.com/Matrices-determinants/4_Multiplying-matrices.php</a> <a href = "http://www.freemathhelp.com/matrix-multiplication.html" target = "_blank">http://www.freemathhelp.com/matrix-multiplication.html</a> <a href = "http://www.sosmath.com/matrix/matrix1/matrix1.html" target = "_blank">http://www.sosmath.com/matrix/matrix1/matrix1.html</a> <a href = "http://joemath.com/applets/multmat/" target = "_blank">http://joemath.com/applets/multmat/</a> REQUIREMENTS When you multiply one matrix by another, you are multiplying the rows of the first matrix by the columns of the second matrix. Because of this, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The third matrix that results from this operation will contain the same number of rows as the first matrix and the same number of columns as the second matrix. EXAMPLE 1: <pre> Rows Columns Matrix 1 2 1 Matrix 2 1 3 Matrix 3 2 3 </pre> Since the number of columns in matrix 1 are the same as the number of rows in matrix 2, then the multiplication can be performed. Matrix 3 will have the same number of rows as matrix 1 and the same number of columns as matrix 2. That would be 2 rows and 3 columns. Matrix 1, 2, and 3 will look like this after the multiplication has been performed. columns are designated as c1, c2, c3, etc. rows are designated as r1, r2, r3, etc. values in each element of each row are designated a, b, c, etc. combined letters within the elements of the matrix indicate multiplication. examples: ac = a*c ad = a*d ae = a*e etc. * is the multiplication symbol <pre> Matrix 1: c1 r1 a r2 b Matrix 2: c1 c2 c3 r1 c d e Matrix 3: c1 c2 c3 r1 ac ad ae r2 bc bd be </pre> row 2 of matrix 1 contains b. column 2 of matrix 2 contains d. row 2 of matrix 1 times column 2 of matrix 2 equals b * d which is shown as bd in row 2 column 2 of matrix 3. EXAMPLE 2: We will now add a column to matrix 1 and a row to matrix 2. <pre> Rows Columns Matrix 1 2 2 Matrix 2 2 3 Matrix 3 2 3 </pre> Since the number of columns in matrix 1 are the same as the number of rows in matrix 2, then the multiplication can be performed. Matrix 3 will have the same number of rows as matrix 1 and the same number of columns as matrix 2. That would be 2 rows and 3 columns. Matrix 1, 2, and 3 will look like this after the multiplication has been performed. <pre> Matrix 1: c1 c2 r1 a f r2 b g Matrix 2: c1 c2 c3 r1 c d e r2 h i j Matrix 3: c1 c2 c3 r1 ac+fh ad+fi ae+fj r2 bc+gh bd+gi be+gj </pre> row 1 of matrix 1 contains a,f respectively reading from left to right. column 3 of matrix 2 contains e,j respectively reading from top to bottom. row 1 of matrix 1 times column 3 of matrix 2 equals a*e + f*j which is shown as ae+fj in row 1 column 3 of matrix 3. If we visually trace the row and column interactions, then the 3 matrices would look like this: <font color = "red"> <img src = "http://theo.x10hosting.com/examples/Matrix/Matrix2.jpg" alt = "***** PICTURE NOT FOUND *****" /> </font> <font color = "red"> <img src = "http://theo.x10hosting.com/examples/Matrix/Matrix3.jpg" alt = "***** PICTURE NOT FOUND *****" /> </font> EXAMPLE 3: We will now add another column to matrix 1 and another row to matrix 2. <pre> Rows Columns Matrix 1 2 3 Matrix 2 3 3 Matrix 3 2 3 </pre> Since the number of columns in matrix 1 are the same as the number of rows in matrix 2, then the multiplication can be performed. Matrix 3 will have the same number of rows as matrix 1 and the same number of columns as matrix 2. That would be 2 rows and 3 columns. Matrix 1, 2, and 3 will look like this after the multiplication has been performed. <pre> Matrix 1: c1 c2 c3 r1 a f k r2 b g l Matrix 2: c1 c2 c3 r1 c d e r2 h i j r3 m n o Matrix 3: c1 c2 c3 r1 ac+fh+km ad+fi+kn ae+fj+ko r2 bc+gh+lm bd+gi+ln be+gj+lo </pre> row 2 of matrix 1 contains b,g,l respectively, reading from left to right. column 1 of matrix number 2 contains c,h,m respectively, reading from top to bottom. row 2 of matrix 1 times column 1 of matrix 2 equals b*c + g*h + l*m which is shown as bc+gh+lm in row 2 column 1 of matrix 3. row 1 of matrix 1 contains a,f,k respectively, reading from left to right. column 3 of matrix 2 contains e,j,o respectively, reading from top to bottom. row 1 of matrix 1 times column 3 of matrix 2 equals a*e + f*j + k*o which is shown as ae+fj+ko in row 1 column 3 of matrix 3. We'll do one more example using letters and then we'll move into numbers. EXAMPLE 4 We will now add another row to matrix 1 and another column to matrix 2. Note that, previously, we were adding a column to matrix 1 and a row to matrix 2. This time we're adding a row to matrix 1 and a column to matrix 2. <pre> Rows Columns Matrix 1 3 3 Matrix 2 3 4 Matrix 3 2 3 </pre> Since the number of columns in matrix 1 are the same as the number of rows in matrix 2, then the multiplication can be performed. Matrix 3 will have the same number of rows as matrix 1 and the same number of columns as matrix 2. That would be 3 rows and 4 columns. Matrix 1, 2, and 3 will look like this after the multiplication has been performed. <pre> Matrix 1: c1 c2 c3 r1 a f k r2 b g l r3 r s t Matrix 2: c1 c2 c3 c4 r1 c d e u r2 h i j v r3 m n o w Matrix 3: c1 c2 c3 c4 r1 ac+fh+km ad+fi+kn ae+fj+ko au+fv+kw r2 bc+gh+lm bd+gi+ln be+gj+lo bu+gv+lw r3 rc+sh+tm rd+si+tn re+sj+to ru+sv+tw </pre> row 3 of matrix 1 contains r,s,t respectively reading from left to right. column 4 of matrix 2 contains u,v,w respectively reading from top to bottom. row 3 of matrix 1 times column 4 of matrix 2 equals r*u + s*v + t*w which is shown as ru+sv+tw in row 3 column 4 of matrix 3. The same principle applies regardless of the number of rows or columns used. EXAMPLE 5 We will now do example 4 using different numbers in each element. <pre> Rows Columns Matrix 1 3 3 Matrix 2 3 4 Matrix 3 2 3 </pre> Since the number of columns in matrix 1 are the same as the number of rows in matrix 2, then the multiplication can be performed. Matrix 3 will have the same number of rows as matrix 1 and the same number of columns as matrix 2. That would be 3 rows and 4 columns. Matrix 1, 2, and 3 will look like this after the multiplication has been performed. <pre> Matrix 1: c1 c2 c3 r1 1 2 3 r2 4 5 6 r3 7 8 9 Matrix 2: c1 c2 c3 c4 r1 10 11 12 13 r2 14 15 16 17 r3 18 19 20 21 Matrix 3: c1 c2 c3 c4 r1 92 98 104 110 r2 218 233 248 263 r3 344 368 392 416 </pre> MATRIX MULTIPLICATION CALCULATOR There is a handy little tool that will help you with matrix multiplication. That tool can be found at the following link: <a href = "http://joemath.com/applets/multmat/" target = "_blank">http://joemath.com/applets/multmat/</a> The output from that tool for the problem example just presented would look like this: <font color = "red"> <img src = "http://theo.x10hosting.com/examples/Matrix/Matrix1.jpg" alt = "***** PICTURE NOT FOUND *****" /> </font>
