2.2E: Matrix Addition, Scalar Multiplication, and Transposition Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
Exercises for 1
Find a, b, c, and d if
- [abcd]=[c−3d−d2a+da+b]
- [a−bb−cc−dd−a]=2[11−31]
- 3[ab]+2[ba]=[12]
- [abcd]=[bcda]
- Answer
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- (a b c d)=(−2,−4,−6,0)+t(1,1,1,1),
- t arbitrary a=b=c=d=t, t arbitrary
Compute the following:
- [321510]−5[30−21−12]
- 3[3−1]−5[62]+7[1−1]
- [−2132]−4[1−20−1]+3[2−3−1−2]
- [3−12]−2[934]+[311−6]
- [1−5402106]T
- [0−1210−4−240]T
- [3−121]−2[1−211]T
- 3[21−10]T−2[1−123]
- Answer
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- [−14−20] (−12,4,−12)
- [01−2−1042−40]
- [4−1−1−6]
Let A=[210−1],
B=[3−12014], C=[3−120],
D=[13−1014], and E=[101010].
Compute the following (where possible).
- 3A−2B
- 5C
- 3ET
- B+D
- 4AT−3C
- (A+C)T
- 2B−3E
- A−D
- ((B - 2E)^{T}\)
- Answer
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- [15−5100]
- Impossible
- [520−1]
- Impossible
Find A if:
- 5A−[1023]=3A−[5261]
- 3A−[21]=5A−2[30]
- Answer
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b. [2−12]
Find A in terms of B if:
- A+B=3A+2B
- 2A−B=5(A+2B)
- Answer
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b. A=−113B
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.
- 5X+3Y=A2X+Y=B
- 4X+3Y=A5X+4Y=B
- Answer
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b. X=4A−3B, Y=4B−5A
Find all matrices X and Y such that:
- 3X−2Y=[3−1]
- 2X−5Y=[12]
Solution
Y=(s,t), X=12(1+5s,2+5t); s and t arbitrary
Simplify the following expressions where A, B, and C are matrices.
- 2[9(A−B)+7(2B−A)]
−2[3(2B+A)−2(A+3B)−5(A+B)] - 5[3(A−B+2C)−2(3C−B)−A]
+2[3(3A−B+C)+2(B−2A)−2C]
- Answer
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b. 20A−7B+2C
If A is any 2×2 matrix, show that:
- A=a[1000]+b[0100]+c[0010]+d[0001] for some numbers a, b, c, and d.
- A=p[1001]+q[1100]+r[1010]+s[0110] for some numbers p, q, r, and s.
- Answer
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b. If A=[abcd], then (p,q,r,s)=12(2d,a+b−c−d,a−b+c−d,−a+b+c+d).
Let A=[11−1],
B=[012], and C=[301]. If
rA+sB+tC=0 for some scalars r, s, and t, show that necessarily r=s=t=0.
- If Q+A=A holds for every m×n matrix A, show that Q=0mn.
- If A is an m×n matrix and A+A′=0mn, show that \(A^\prime = -A\
- Answer
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b. If A+A′=0 then −A=−A+0=−A+(A+A′)=(−A+A)+A′=0+A′=A′
If A denotes an m×n matrix, show that A=−A if and only if A=0.
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.
- A+B
- A−B
- kA for any number k
- Answer
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Write A=\diag(a1,…,an), where a1,…,an are the main diagonal entries. If B=\diag(b1,…,bn) then kA=\diag(ka1,…,kan).
In each case determine all s and t such that the given matrix is symmetric:
- [1s−2t]
- [stst1]
- [s2sstt−1sts2s]
- [2st2s0s+t33t]
- Answer
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- s=1 or t=0
- s=0, and t=3
In each case find the matrix A.
- (A+3[1−10124])T=[210538]
- (3AT+2[1002])T=[8031]
- (2A−3[120])T=3AT+[21−1]T
- (2AT−5[10−12])T=4A−9[11−10]
- Answer
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- [201−1]
- [27−92−5]
Let A and B be symmetric (of the same size). Show that each of the following is symmetric.
- (A−B)
- kA for any scalar k
- Answer
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- A=AT,
- (kA)T=kAT=kA.
Show that A+AT is symmetric for any square matrix A.
If A is a square matrix and A=kAT where k≠±1, show that A=0.
In each case either show that the statement is true or give an example showing it is false.
- If A+B=A+C, then B and C have the same size.
- If A+B=0, then B=0.
- If the (3,1)-entry of A is 5, then the (1,3)-entry of AT is −5.
- A and AT have the same main diagonal for every matrix A.
- If B is symmetric and AT=3B, then A=3B.
- If A and B are symmetric, then kA+mB is symmetric for any scalars k and m.
- Answer
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- False. Take B=−A for any A≠0.
- True. Transposing fixes the main diagonal.
- True. (kA+mB)T=(kA)T+(mB)T=kAT+mBT=kA+mB
A square matrix W is called skew-symmetric if WT=−W. Let A be any square matrix.
- Show that A−AT is skew-symmetric.
- Find a symmetric matrix S and a skew-symmetric matrix W such that A=S+W.
- Show that S and W in part (b) are uniquely determined by A.
- Answer
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c. Suppose A=S+W, where S=ST and W=−WT. Then AT=ST+WT=S−W, so A+AT=2S and A−AT=2W. Hence S=12(A+AT) and W=12(A−AT) are uniquely determined by A.
If W is skew-symmetric (Exercise ), show that the entries on the main diagonal are zero.
Prove the following parts of Theorem [thm:002170].
- (k+p)A=kA+pA
- (kp)A=k(pA)
- Answer
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b. If A=[aij] then (kp)A=[(kp)aij]=[k(paij)]=k[paij]=k(pA).
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.
- (k(A_{1} + A_{2} + \dots + A_{n}) = kA_{1} + kA_{2} + \dots + kA_{n}\) for any number k
- (k1+k2+⋯+kn)A=k1A+k2A+⋯+knA for any numbers k1,k2,…,kn
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.