By N. Balabanian; T.A. Bickart; S. Seshu
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Furthermore, A is the similarity transform of B by S . The reason that similarity of matrices is an important concept is the fact that similar matrices have equal determinants, the same character istic polynomials, and, hence, the same eigenvalues. These facts are easily established. Thus, by applying the rule for the determinant of a product of square matrices, the determinants are equal, because - 1 -1 - 1 - 1 B| = | S A S | = | S | |A| |S| = | S S | |A| = |A|. The characteristic polynomials are equal because - 1 -1 |λU - B| = |λU - S A S | = | S ( λ U - A)S| - 1 - 1 = | S | |λU - A| |S| = | S S | |λU - A| = |λU-A|.
If every element of any row or any column is zero, then the deter minant is zero. 6. The determinant is unchanged if to each element of any row or column is added a scalar multiple of the corresponding element of any other row or column. Cofactor Expansion. Let A be a square matrix of order n. If the ith row and jth column of A are deleted, the determinant of the remaining Sec. 2] ELEMENTARY MATRIX ALGEBRA 13 matrix of order n — 1 is called a first minor (or simply a minor) of A or of det A and is denoted by M .
The reciprocal of L has no name, but t h e symbol Γ (an inverted L) is sometimes used. Finally, t h e capacitor is described by t h e capacitance parameter C. The reciprocal of C is given t h e name elastance, and t h e symbol D is some times used. A number of comments are in order concerning these elements.