Matrix Algebra and Matrix Multiplication - Revision history

Lila: /* Resources */

2021-11-03T13:30:17Z

Resources

2021-11-03T13:29:22Z

Resources

2021-09-17T18:00:46Z

2021-09-17T17:57:36Z

Multiplying matrices

2021-09-17T17:54:46Z

Multiplying matrices

2021-09-17T17:53:19Z

Multiplying matrices

2021-09-17T17:50:27Z

Multiplying matrices

2021-09-17T17:49:40Z

Multiplying matrices

2021-09-17T17:45:24Z

2021-09-17T17:43:38Z

Multiplying matrices

← Older revision		Revision as of 13:30, 3 November 2021
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	* [https://openstax.org/books/college-algebra/pages/7-5-matrices-and-matrix-operations Matrices and Matrix Operations], OpenStax		* [https://openstax.org/books/college-algebra/pages/7-5-matrices-and-matrix-operations Matrices and Matrix Operations], OpenStax
	* [https://www.youtube.com/watch?v=0L90Kkn90J8 Multiplying a 2-by-3 Matrix and 3-by-2 Matrix], patrickJMT		* [https://www.youtube.com/watch?v=0L90Kkn90J8 Multiplying a 2-by-3 Matrix and 3-by-2 Matrix], patrickJMT
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Linear_Algebra/Inverses Inverses, Wikibooks: Linear Algebra] under a CC BY-SA license
		+	* [https://en.wikibooks.org/wiki/Linear_Algebra/The_Inverse_of_a_Matrix The Inverse of a Matrix, Wikibooks: Linear Algebra] under a CC BY-SA license

@@ Line 118: / Line 118: @@
 </math>
-Note that there is no "matrix division". Rather, we multiply one matrix by the inverse of another to obtain a "quotient" from two matrices.
+Note that there is no "matrix division". Rather, we multiply one matrix by the inverse of another (for example, <math> AB^{-1} </math>, NOT <math> A/B </math>) to obtain a "quotient" from two matrices.
 ==Resources==
@@ Line 124: / Line 124: @@
 * [https://www.khanacademy.org/math/algebra-home/alg-matrices/alg-multiplying-matrices-by-scalars/v/scalar-multiplication Multiplying Matrices by Scalars], Khan Academy
 * [https://www.khanacademy.org/math/algebra-home/alg-matrices/alg-multiplying-matrices-by-matrices/v/matrix-multiplication-intro Matrix Multiplication], Khan Academy

@@ Line 117: / Line 117: @@
 \end{bmatrix}
 </math>
 ==Resources==

@@ Line 107: / Line 107: @@
 (5) + 2(7) & 1(6) + 2(8)\\
 (5) + 4(7) & 3(6) + 4(8)
 \end{bmatrix}
 </math>

← Older revision		Revision as of 17:53, 17 September 2021
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	where <math>c_{ij}</math> is the dot product of the i-th row of the first matrix and the j-th column of the second matrix; that is, <math>c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + a_{i3}b_{3j}</math>.		where <math>c_{ij}</math> is the dot product of the i-th row of the first matrix and the j-th column of the second matrix; that is, <math>c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + a_{i3}b_{3j}</math>.
		+
		+	Here is a 2-by-2 example of matrix multiplication:
		+
		+	<math>\begin{bmatrix}
		+	1 & 2\\
		+	3 & 4
		+	\end{bmatrix}
		+	\begin{bmatrix}
		+	5 & 6\\
		+	7 & 8
		+	\end{bmatrix} =
		+	\begin{bmatrix}
		+	1(5) + 2(7) & 1(6) + 2(8)\\
		+	3(5) + 4(7) & 3(6) + 4(8)
		+	\end{bmatrix}
		+	</math>

	==Resources==		==Resources==

@@ Line 92: / Line 92: @@
 </math>
-where c_{ij} is the dot product of the i-th row of the first matrix and the j-th column of the second matrix; that is, c_{ij} = a_{i1}b_{1j} & a_{i2}b_{2j} & a_{i3}b_{3j}.
+where <math>c_{ij}</math> is the dot product of the i-th row of the first matrix and the j-th column of the second matrix; that is, <math>c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + a_{i3}b_{3j}</math>.
 ==Resources==

@@ Line 72: / Line 72: @@
 ===Multiplying matrices===
-Matrix multiplication is not commutative; that is, for most matrices <math> A </math> and <math> B </math>, <math> AB \neq BA </math>. Two matrices can only be multiplied together if the number of columns in the first matrix equals the number of rows in the second matrix. For example, we can take the product of a 3-by-2 matrix times a 2-by-5 matrix, but not a 3-by-2 matrix times a 3-by-2 matrix.
+Matrix multiplication is not commutative; that is, for most matrices <math> A </math> and <math> B </math>, <math> AB \neq BA </math>. Two matrices can only be multiplied together if the number of columns in the first matrix equals the number of rows in the second matrix. For example, we can take the product of a 3-by-2 matrix times a 2-by-5 matrix, but not a 3-by-2 matrix times a 3-by-2 matrix. The product of two matrices has the same number of rows as the first matrix and the same number of columns as the second matrix. For example, a 2-by-3 matrix times a 3-by-2 matrix will result in a 2-by-2 matrix.
 <math>\begin{bmatrix}
 a_{11} & a_{12} & a_{13}\\
@@ Line 90: / Line 91: @@
 \end{bmatrix}
 </math>
 ==Resources==