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Question feed. Finally, taking the dot product of row 2 in A with columns 3 and 4 in B gives respectively the 2, 3 and 2, 4 entries in AB :. First, note that since C is 4 x 5 and D is 5 x 6, the product CD is indeed defined, and its size is 4 x 6. The 3, 5 entry of CD is the dot product of row 3 in C and column 5 in D :. The previous example gives one illustration of what is perhaps the most important distinction between the multiplication of scalars and the multiplication of matrices.
However, it is decidedly false that matrix multiplication is commutative. In fact, the matrix AB was 2 x 2, while the matrix BA was 3 x 3. Here is another illustration of the noncommutativity of matrix multiplication: Consider the matrices. Since C is 3 x 2 and D is 2 x 2, the product CD is defined, its size is 3 x 2, and.
The product DC , however, is not defined, since the number of columns of D which is 2 does not equal the number of rows of C which is 3. Since A is 2 x 2, in order to multiply A on the right by a matrix, that matrix must have 2 rows. Therefore, if x is written as the 2 x 1 column matrix. Show that any two square diagonal matrices of order 2 commute. Although matrix multiplication is usually not commutative, it is sometimes commutative; for example, if. Despite examples such as these, it must be stated that in general, matrix multiplication is not commutative.
There is another difference between the multiplication of scalars and the multiplication of matrices. That is, the only way a product of real numbers can equal 0 is if at least one of the factors is itself 0. The analogous statement for matrices, however, is not true. For instance, if.
Note that even though neither G nor H is a zero matrix, the product GH is. Yet another difference between the multiplication of scalars and the multiplication of matrices is the lack of a general cancellation law for matrix multiplication. For example, if. Example 13 : Although matrix multiplication is not always commutative, it is always associative.
That is, as long as the order of the factors is unchanged, how they are grouped is irrelevant. Note that the associative law implies that the product of A, B , and C in that order can be written simply as ABC ; parentheses are not needed to resolve any ambiguity, because there is no ambiguity. In fact, the equation. This says that if the product AB is defined, then the transpose of the product is equal to the product of the transposes in the reverse order.
Identity matrices. The zero matrix 0 m x n plays the role of the additive identity in the set of m x n matrices in the same way that the number 0 does in the set of real numbers recall Example 7. Similarly, the matrix. In general, the matrix I n —the n x n diagonal matrix with every diagonal entry equal to 1—is called the identity matrix of order n and serves as the multiplicative identity in the set of all n x n matrices.
If A is the matrix. Like A , the matrix B must be 2 x 2. This equation proves that A 2 will commute with A for any square matrix A ; furthermore, it suggests how one can prove that every integral power of a square matrix A will commute with A.
This matrix B does indeed commute with A , as verified by the calculations. However, to establish that the formula holds for all positive integers n , a general proof must be given.
This will be done here using the principle of mathematical induction , which reads as follows. Let P n denote a proposition concerning a positive integer n. If it can be shown that. In the present case, the statement P n is the assertion. By the principle of mathematical induction, the proof is complete.
The inverse of a matrix. Let a be a given real number. Since 1 is the multiplicative identity in the set of real numbers, if a number b exists such that. The analog of this statement for square matrices reads as follows.
Let A be a given n x n matrix. Yet another distinction between the multiplication of scalars and the multiplication of matrices is provided by the existence of inverses. Although every nonzero real number has an inverse, there exist nonzero matrices that have no inverse. However, since the second row of A is a zero row, you can see that the second row of the product must also be a zero row:. Since the 2, 2 entry of the product cannot equal 1, the product cannot equal the identity matrix.
Therefore, it is impossible to construct a matrix that can serve as the inverse for A. If a matrix has an inverse, it is said to be invertible.
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