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본 자료는 교재의 저자인 곽진호 교수님의 협조와 지원으로 성대에서의 강의 기록을 기초로 위의 교재를 이용하시는 강의자 여러분들과 교수 학습 자료를 공유한다는 의미에서 정리한 것입니다. 

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이상구교수의 2003년 1학기 행렬론 문제와 답안 파일을 기초로

2004년 버전 교재의 문제 번호와 순서대로 다시 정리한 답안입니다.

1.1 Which of the following matrices are in row-echelon form or in reduced row-echelon form?

1.2 Find a row-echelon form of each matrix.

(1) (2)

1.4 Solve the systems of equations by Gauss-Jordan elimination. What are the pivots in each elimination step?

(1)   

(2) 

1.5 Which of the following systems has a nontrivial solution?

(1)       (2)

1.6 Determine all values of the that make the following system consistent:

1.7 Determine the condition on so that the following system has no solution:

1.8  Let and be matrices of the same size.

(1) Show that, if for all , then is the zero matrix.

(2) Show that, if for all , then .

1.9 Compute and for

1.10 Prove that if is a 3×3 matrix such that for every 3×3 matrix , then for some constant .

1.11 Let .  Find for all integers .

1.12 Compute and for

1.13 Let be a polynomial.

For any square matrix , a matrix polynomial is defined as .

For ,  find for

(1) (2)

1.14 Find the symmetric part and the skew-symmetric part of each of the following matrices.

(1) (2)

1.15 Find and for the matrix  .

1.16 Let

(1) Find a matrix such that

(2) Find a matrix such that

1.17 Find all possible choices of and so that has an inverse matrix such that

1.18 Decide wether or not each of the following matrices is invertible. Find the inverses for invertible ones.

1.19 Suppose is a matrix and is a matrix. Prove that is not invertible.

1.20 Find three matrices which are row equivalent to .

1.22 Write the following systems of equations as matrix equations and solve them by computing;

(1)       (2)

1.23 Find the LDU factorization for each of the following matrices

(1) (2)

1.24 Find the factorization of the following symmetric matrices.

(1)

(2) If and

    로 factorization을 하려면 이어야 한다 (i.e )

1.25 Solve with , where and are given as

    ,   ,    ,

    Forward elimination is the same as and back-substitution is .

1.26  Let and .

(1) Solve by the Gauss-Jordan elimination.

(2) Find the factorization of .

(3) Write as a product of elementary matrices.

(4) Find the inverse of .

1.27 A square matrix is said to be nilpotent if for a positive integer .

(1) Show that an invertible matrix is not nilpotent.

(2) Show that any triangular matrix with zero diagonal is nilpotent.

(3) Show that if is a nilpotent with = 0, then is invertible with its inverse .

1.28. A square matrix is said to be idempotent if

(1) Find an example of an idempotent matrix other than or .

(2) Show that, if a matrix is both idempotent and invertible, then .

1.29. Determine whether the following statements are true or false, in general, and justify your answers.

(1) Let and be row-equivalent square matrices. Then is nonsingular if and only if is non-singular.

(2) Let be a square matrix such that , then is the identity.

(3) If and are non-singular matrices such that and , then  

(4) If and are non-singular matrices, is also non-singular. 

(5) If and are symmetric, then .

(6) If and are symmetric of the same size, then is also symmetric.

(7) If is invertible and symmetric, then is also symmetric.

(8) Let . Then is invertible if and only if is invertible.

(9) If a square matrix is not invertible, then neither is any for .

(10) If and are elementary matrices, then .

(11) The inverse of an invertible upper triangular matrix is upper triangular.

(12) Any invertible martrix can be written as , where is lower triangular and is upper triangular.

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