Begin2.DVI

The transpose matrix The transpose of a matrix A= (aij)m×nis obtained by interchanging the rows and columns of the matrix A. The ...

A=   3 2 1 1 0 2 4 − 6 0 0 3 1 0 0 0 2   Figure 10-2. A 4 × 4 upper triangular matrix. Diagonal matrices A matrix which ha ...

The trace of a matrix The trace of a n×nsquare matrix A is denoted Tr(A)and represents a summation of the diagonal elements of t ...

Example 10-4. For Aan n×nsquare matrix, show that (A−^1 )−^1 =A. That is, show the inverse of an inverse matrix is again the ori ...

Matrices with Special Properties The following is some terminology associated with square matrices Aand B. (1) If AB =−BA, then ...

An orthogonal matrix If A is an n×nsquare matrix satisfying ATA=AA T =I, then Ais called an orthogonal matrix , and A−^1 =AT.An ...

Example 10-10. Consider the fixed set of axes x, y and a set of barred axes x, ̄y ̄where the barred set of axes is rotated about ...

Example 10-11. Represent the given system of differential equations in matrix form. dy 1 dt =y 1 +y 2 −y 3 + sin t, dy^2 dt = 2y ...

Example 10-13. Represent the differential equation d (^2) y dt^2 +ω^2 y= sin 2 tin matrix form. Solution Let y 1 =yand y 2 = dy ...

This implies that the matrix product ZX =C is a constant. If the matrix X is nonsingular, then X−^1 exists, so that one can solv ...

principle of multiplication by the number of distinct ways a thing can be done can be extended to more than just two or three so ...

Here (1 ,2) is an even permutation of (1 ,2) and (2 ,1) represents an odd permuta- tion of (1 ,2) A mnemonic device to remember ...

Solution The definition of a determinant gives the relation y=y(t) = ∑ (−1)mai 1 (t)aj 2 (t) where the summation is over all per ...

|A|= det A=a 11 ∣∣ ∣∣a^22 a^23 a 32 a 33 ∣∣ ∣∣−a 12 ∣∣ ∣∣a^21 a^23 a 32 a 33 ∣∣ ∣∣+a 13 ∣∣ ∣∣a^21 a^22 a 31 a 32 ∣∣ ∣∣=a 11 c 11 ...

∣∣ ∣∣ ∣∣ ∣∣ ∣∣ ∣ a 11 a 12 ··· a 1 n .. . .. . ... .. . qa i 1 qa i 2 ··· qa in .. . .. . ... .. . an 1 an 2 ··· ann ∣∣ ∣∣ ∣∣ ∣∣ ...

|A|= (1)(−5) + (0)(5) + (1)(−5) = − 10 |A|= (−1)(2) + (1)(−4) + (2)(−2) = − 10 |A|= (3)(−1) + (2)(−3) + (−1)(1) = − 10 and using ...

multiply row 1 by two and add the result to row 3, and (c) subtract row 1 from row 2. Performing these calculations produces |A| ...

The determinant of Ais then obtained by multiplying all the elements on the main diagonal and |A|= det(A) = a 11 a 22... ann = ∏ ...

AdjA=CT= [ 4 − 2 3 1 ] . This gives |A|= 10 so that Ais nonsingular and the inverse is given by A−^1 = 1 10 [ 4 − 2 3 1 ] As a c ...

Observe that multiplication of the matrix Aby an elementary matrix E produces the following elementary transformations of the ma ...