Matrix Multiplication Rules Pdf
Example 7 A B. In this case it is the element of M nm whose ijth entry is given by BA ij A i1B 1j A i2B 2j A ipB pj.
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Matrix multiplication not commutative In general AB BA.
Matrix multiplication rules pdf. We see that the elements of 2A are each twice the elements of A. Perform translation then rotation 0 M Identity 1 translation Ttx ty 01 translation Ttxty0 -MMxTtxty0 M M x Ttxty0 2 rotation R - M M x R 3 Now transform a point P - P M x P. Multiplication of a multidimensional matrix by a scalar results in multiplying every element of the multidimensional matrix by the scalar.
Matrix A we find AA 4 3 0 1. BA 3 4 3 1 2 2 7 11 9 4 3 2 5 6 3 3 5 2 1 0 0 0 1 0 0 0 1 I. The resulting matrix known as the matrix product has the number of rows of the first and the number of columns of the second matrix.
An important observation about matrix multiplication is related to ideas from vector spaces. The following task shows that this is not necessarily true for matrices. Then B 1 2 B BT 1 2 B BT.
Problems with hoping AB and BA are equal. For matrix multiplication the number of columns in the first matrix must be equal to the number of rows in the second matrix. Now in the multiplication of numbers the equation ab 0 implies that either a is zero or b is zero or both are zero.
We express them in matrix form. Even if AB and BA are both defined BA may not be the same size. - We can get the negative of a matrix by using the above multiplication method.
Hence both are the zero matrix. When the number of columns of A equals the number of rows of B the two matrices are said to be conformable and the product AB is obtained as follows. 1 7 3 1 1 1 1 3 1 2 1 1 3 2 1 x x x Where matrix A is 1 1 1 1 3 1 2 1 1 A and vector y is 1 7 3 According to Cramers rule.
Carry out the multiplication AB where A 1 1 1 1 B 1 1 1 1 Your solution. The multiplication can be performed and the result will be a 2 2 matrix. The matrix B is the inverse of the matrix A and this is usually written as A1.
Example 6 PART D - Multiplying Matrices We can multiply a matrix A by another matrix B if the number of columns in A is equal to the number of rows in B in bold. The left matrix is symmetric while the right matrix is skew-symmetric. 1 3 1 1 7 3 1 1 1 1 8 2 4 x A To find x1 we replace the first column of A with vector y and divide the determinant of this new matrix by the determinant of A.
Theorem 3 Algebraic Properties of Matrix Multiplication 1. KA B kA kB Distributivity of scalar multiplication II 3. K A kA A Distributivity of scalar multiplication I 2.
If the two middle numbers dont match you cant multiply the matrices. A J 0 1 10 o is skew-symmetric. 44 33 00 11.
Combination If Eis an elementary matrix for a combination rule then detEA detA. Matrix-vectorproduct very important special case of matrix multiplication. BA may not be well-defined.
2 4 5 3. In mathematics particularly in linear algebra matrix multiplication is a binary operation that produces a matrix from two matrices. OpenGL Post-Multiplication OpenGL post-multiplies each new transformation matrix M M x M new Example.
Equally the matrix A is the inverse of the matrix B. The first matrix has size 22. One of the most important rules regarding matrix multiplication is the following.
Let B 12 14 BT 1 1 24 B BT 03 30 B BT 21 18. If m 1 multiplication by B is a map Rp M p1 R n M n1. ABC ABCAssociativity of matrix mul-tiplication 5.
It sends a column vector X x 1 x p to BX x 1C 1Bx 2C. Clearly the number of columns in the first is the same as the number of rows in the second. The second matrix has size 22.
This illustrates how to multiply a matrix by a number and leads us to the topic of scalarmultiplication. 4 3 0 1. Another way to write AA is 2A.
Y Ax A is an mn matrix x is an n-vector y is an m-vector y i A i1x1A inx n i 1m can think of y Ax as a function that transforms n-vectors into m-vectors a set of m linear equations relating x to y Matrix Operations 29. Eg A is 2 x 3 matrix B is 3 x 5 matrix eg A is 2 x 3 matrix B is 3 x 2 matrix. In the following example a 4-D matrix with dimensions of 3 2 1 2 is multiplied by the scalar 5 with the resulting 4-D matrix with dimensions of 3 2 1 2 as shown.
ABC ABACDistributivity of matrix multiplication 4. We call this associativity and that matrix multiplication. Therefore we have 2A 8 6 0 2.
Thus the matrix product is an operation. 2 48 12 57. Multiply If Eis an elementary matrix for a multiply rule with multiplier c6 0 then detEA cdetA.
Some of these are1 ABBA cABcAcB ABCABC CAB CACB ABCACBC ABC ABC Perhaps the most interesting and unexpected of the above rules is ABG ABC. 2 2 3 1 1 7 1 1 1 1 4 1 4 x A. Multiplication of A by B is typically written as AB or AB.
Pm the matrix product BA is defined if q p. 3 6 1 9. Solution Using the rules of matrix multiplication AB 4 3 2 5 6 3 3 5 2 3 4 3 1 2 2 7 11 9 1 0 0 0 1 0 0 0 1 I.
A 1 2 AAT 1 2 AAT. Even if AB and BA are both defined and of the same size they still may not be equal. We call B the 22 zero matrix written 0 so that A0 0A 0 for any matrix A.
5 1 4 7 10 2 5 8 11 3 6 9 12. 2 34 1 2 64 9 5 33 1 5 639. 8 6 0 2.
M np M pm M nm. 2 Laws of Matrix Arithmetic Many of the standard rules from ordinary arithmetic carry over into matrix arithmetic. A B B ACommutativity of matrix ad-dition 6.
Since detE 1 for a combination rule detE 1 for a swap rule and detE cfor a multiply rule.
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