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augmented matrix bilinear form C_1AC characteristic equation characteristic polynomial coefficient matrix column vectors congruent Cramer's rule defined Determine the rank diagonal matrix elementary matrices elementary row elementary transformations equations in Eq Example Exercises exists a nonsingular expanded Find the characteristic following matrices following theorem hence homogeneous equations integer inverse linear combination linear equations linear homogeneous linear space linear vector space linearly independent vectors matrix of order minimum polynomial multiplication mutually orthogonal n-dimensional vectors non-zero nonsingular matrix normal form obtained orthogonal matrix P'AP positive definite matrix postmultiplying premultiplying principal diagonal principal minors product of elementary Prove the theorem quadratic form real numbers real symmetric matrix result row column row vectors Section set of vectors Show similar matrices singular solution of Eq square matrix submatrices symmetric matrix system of equations transpose triangular matrix unit vectors unitary matrix Verify write x'Ax zero vector