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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.

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