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Mathematics LibreTexts

2.2E: Matrix Addition, Scalar Multiplication, and Transposition Exercises

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Exercises for 1

Exercise 2.2E.1

Find a, b, c, and d if

  1. [abcd]=[c3dd2a+da+b]
  2. [abbccdda]=2[1131]
  3. 3[ab]+2[ba]=[12]
  4. [abcd]=[bcda]
Answer
  1. (a b c d)=(2,4,6,0)+t(1,1,1,1),
  2. t arbitrary a=b=c=d=t, t arbitrary
Exercise 2.2E.2

Compute the following:

  1. [321510]5[302112]
  2. 3[31]5[62]+7[11]
  3. [2132]4[1201]+3[2312]
  4. [312]2[934]+[3116]
  5. [15402106]T
  6. [012104240]T
  7. [3121]2[1211]T
  8. 3[2110]T2[1123]
Answer
  1. [1420] (12,4,12)
  2. [012104240]
  3. [4116]
Exercise 2.2E.3

Let A=[2101],
B=[312014], C=[3120],
D=[131014], and E=[101010].

Compute the following (where possible).

  1. 3A2B
  2. 5C
  3. 3ET
  4. B+D
  5. 4AT3C
  6. (A+C)T
  7. 2B3E
  8. AD
  9. ((B - 2E)^{T}\)
Answer
  1. [155100]
  2. Impossible
  3. [5201]
  4. Impossible
Exercise 2.2E.4

Find A if:

  1. 5A[1023]=3A[5261]
  2. 3A[21]=5A2[30]
Answer

b. [212]

Exercise 2.2E.5

Find A in terms of B if:

  1. A+B=3A+2B
  2. 2AB=5(A+2B)
Answer

b. A=113B

Exercise 2.2E.6

If X, Y, A, and B are matrices of the same size, solve the following systems of equations to obtain X and Y in terms of A and B.

  1. 5X+3Y=A2X+Y=B
  2. 4X+3Y=A5X+4Y=B
Answer

b. X=4A3B, Y=4B5A

Example 2.2E.7

Find all matrices X and Y such that:

  1. 3X2Y=[31]
  2. 2X5Y=[12]
Solution

Y=(s,t), X=12(1+5s,2+5t); s and t arbitrary

Exercise 2.2E.8

Simplify the following expressions where A, B, and C are matrices.

  1. 2[9(AB)+7(2BA)]
    2[3(2B+A)2(A+3B)5(A+B)]
  2. 5[3(AB+2C)2(3CB)A]
    +2[3(3AB+C)+2(B2A)2C]
Answer

b. 20A7B+2C

Exercise 2.2E.9

If A is any 2×2 matrix, show that:

  1. A=a[1000]+b[0100]+c[0010]+d[0001] for some numbers a, b, c, and d.
  2. A=p[1001]+q[1100]+r[1010]+s[0110] for some numbers p, q, r, and s.
Answer

b. If A=[abcd], then (p,q,r,s)=12(2d,a+bcd,ab+cd,a+b+c+d).

Exercise 2.2E.10

Let A=[111],
B=[012], and C=[301]. If
rA+sB+tC=0 for some scalars r, s, and t, show that necessarily r=s=t=0.

Exercise 2.2E.11
  1. If Q+A=A holds for every m×n matrix A, show that Q=0mn.
  2. If A is an m×n matrix and A+A=0mn, show that \(A^\prime = -A\
Answer

b. If A+A=0 then A=A+0=A+(A+A)=(A+A)+A=0+A=A

Exercise 2.2E.12

If A denotes an m×n matrix, show that A=A if and only if A=0.

Exercise 2.2E.13

A square matrix is called a diagonal matrix if all the entries off the main diagonal are zero. If A and B are diagonal matrices, show that the following matrices are also diagonal.

  1. A+B
  2. AB
  3. kA for any number k
Answer

Write A=\diag(a1,,an), where a1,,an are the main diagonal entries. If B=\diag(b1,,bn) then kA=\diag(ka1,,kan).

Exercise 2.2E.14

In each case determine all s and t such that the given matrix is symmetric:

  1. [1s2t]
  2. [stst1]
  3. [s2sstt1sts2s]
  4. [2st2s0s+t33t]
Answer
  1. s=1 or t=0
  2. s=0, and t=3
Exercise 2.2E.15

In each case find the matrix A.

  1. (A+3[110124])T=[210538]
  2. (3AT+2[1002])T=[8031]
  3. (2A3[120])T=3AT+[211]T
  4. (2AT5[1012])T=4A9[1110]
Answer
  1. [2011]
  2. [27925]
Exercise 2.2E.16

Let A and B be symmetric (of the same size). Show that each of the following is symmetric.

  1. (AB)
  2. kA for any scalar k
Answer
  1. A=AT,
  2. (kA)T=kAT=kA.
Exercise 2.2E.17

Show that A+AT is symmetric for any square matrix A.

Exercise 2.2E.18

If A is a square matrix and A=kAT where k±1, show that A=0.

Exercise 2.2E.19

In each case either show that the statement is true or give an example showing it is false.

  1. If A+B=A+C, then B and C have the same size.
  2. If A+B=0, then B=0.
  3. If the (3,1)-entry of A is 5, then the (1,3)-entry of AT is 5.
  4. A and AT have the same main diagonal for every matrix A.
  5. If B is symmetric and AT=3B, then A=3B.
  6. If A and B are symmetric, then kA+mB is symmetric for any scalars k and m.
Answer
  1. False. Take B=A for any A0.
  2. True. Transposing fixes the main diagonal.
  3. True. (kA+mB)T=(kA)T+(mB)T=kAT+mBT=kA+mB
Exercise 2.2E.20

A square matrix W is called skew-symmetric if WT=W. Let A be any square matrix.

  1. Show that AAT is skew-symmetric.
  2. Find a symmetric matrix S and a skew-symmetric matrix W such that A=S+W.
  3. Show that S and W in part (b) are uniquely determined by A.
Answer

c. Suppose A=S+W, where S=ST and W=WT. Then AT=ST+WT=SW, so A+AT=2S and AAT=2W. Hence S=12(A+AT) and W=12(AAT) are uniquely determined by A.

Exercise 2.2E.21

If W is skew-symmetric (Exercise 2.2E.20), show that the entries on the main diagonal are zero.

Exercise 2.2E.22

Prove the following parts of Theorem [thm:002170].

  1. (k+p)A=kA+pA
  2. (kp)A=k(pA)
Answer

b. If A=[aij] then (kp)A=[(kp)aij]=[k(paij)]=k[paij]=k(pA).

Exercise 2.2E.23

Let A,A1,A2,,An denote matrices of the same size. Use induction on n to verify the following extensions of properties 5 and 6 of Theorem 2.2E.1.

  1. (k(A_{1} + A_{2} + \dots + A_{n}) = kA_{1} + kA_{2} + \dots + kA_{n}\) for any number k
  2. (k1+k2++kn)A=k1A+k2A++knA for any numbers k1,k2,,kn
Example 2.2E.1

Let A be a square matrix. If A=pBT and B=qAT for some matrix B and numbers p and q, show that either A=0=B or pq=1.


2.2E: Matrix Addition, Scalar Multiplication, and Transposition Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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