1.The curve shown in the accompanying figure passes through the points , , and . Show that the coefficients a, b, and c are a solution of the system of linear equations whose augmented matrix is
2.Consider the system of equations
Show that for this system to be consistent, the constants a, b, and c must satisfy a = b + c.
3.Solve each of the following systems by Gauss-Jordan elimination.
4.Solve each of the following systems by Gaussian elimination.
5.For which values of a will the following system have no solutions? Exactly one solution? Infinitely many solutions?
6.Find the coefficients a, b, c, and d so that the curve shown in the accompanying figure is the graph of the equation y = ax3 + bx2 + cx + d .
7.Find the 4 × 4 matrix A = [aij] whose entries satisfy the stated condition.
(a) aij = (b) aij = (c) aij =
8.Consider the matrices
, , , ,
Compute the following:
(a) 5E – 2D(b) –3(D+3E)(c) 4 tr(6B)(d) 3AT + C(e)(2ET –DT)T
(f) (4E)D(g) (AB)C(h) (CTB)AT (i) tr(CTAT + 3ET)
(j) (BAT – 3C)T (k) BT(CCT – ATA)
, , , a = 4, b = –7
(a) (AB)C = A(BC)(b) a(B – C) = aB – aC(c) A(B – C) = AB – AC
(d) (A + B)T = AT + BT (e) (AB)T = BTAT(f) (BT)-1 = (B-1)T
(g) (AB)-1 = B-1 A-1
10.Let A be the matrix
In each part find p(A).
(a) p(x) = x – 3(b) p(x) = 2x2 – 2x + 1(c) p(x) = x3 – 2x + 5
1.A square matrix A is called symmetric if AT = A and skew-symmetric if AT = –A. Show that if B is a square matrix, then
(a) BBT and B + BT are symmetric(b) B – BT is skew-symmetric.
2.Show that if a square matrix A satisfies 3A2 – 2A + I = 0, then A-1 = 2I – 3A.
3.Consider the matrix
(a)Find elementary matrices E1 and E2 such that E2E1A = I.
(b)Write A-1 as a product of two elementary matrices.
(c)Write A as a product of two elementary matrices.
4.Prove that if A is an invertible matrix and B is row equivalent to A, then B is also invertible.
5.(a) Prove: If A and B are m × n matrices, then A and B are row equivalent if and only if A and B have the same reduced row-echelon form.
(b) Show that A and B are row equivalent, and find a sequence of elementary row operations that produces B from A.
6.Solve the following matrix equation for X.
7.Let Ax = 0 be a homogeneous system of n linear equations in n unknowns, and let Q be an invertible n × n matrix. Show that Ax = 0 has just the trivial solution if and only if (QA)x = 0 has just the trivial solution.
8.Let A be a square matrix. Show that if An+1 = 0.
9.Prove: If B is invertible, then AB-1 = B-1A if and only if AB = BA.
10.Assuming that the stated inverses exist, prove the following equalities.
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1.The 曲線被顯示在上伴隨的圖通過點, 和。表示, 系數a 、b, 和c 是線性方程的系統的解答被增添的矩陣是圖1 2.Consider 等式系統表示, 使這個系統是一致的, 常數a, b, 和c 必須滿意a = b + c. 3.Solve 每個以下系統由高斯喬丹排除。(a) (b) 4.Solve 每個以下系統由Gaussian 排除。(a) a 的價值以下系統不會有解答的(b) 5.For? 確切一種解答? 無限釵h解答? 6.Find 系數a 、b 、c, 和d 以便曲線被顯示在上伴隨的圖是等式的圖表y = ax3 + bx2 + cx + d 。圖2 7.Find 詞條滿足陳述的情況的4 .6N 4 矩陣A = [ aij ] 。(a) aij = (b) aij = (c) aij = 8.Consider 矩陣,, 計算以下: (a) 5.E .V 2D(b) .V3(D+3.E)(c) 4 tr(6B)(d) 3.AT + C(e)(2.ET .VDT)T (f) (4.E)D(g) (AB)C(h) (CTB)AT (i) tr(CTAT + 3.ET) (j) (棒.V 3C)T (k) BT(CCT .V ATA) 9.Let,, a = 4, b = .V7 展示(a) (AB)C = A(BC)(b) a(B .V C) = aB .V aC(c) A(B .V C) = AB .V AC (d) (A + B)T = 在+ BT (e) (AB)T = BTAT(f) (BT)-1 = (B-1)T (g) (AB)-1 = B-1 A-1 10.Let A 是矩陣在各部份發現p(A) 。(a) p(x) = x .V 3(b) p(x) = 2x2 .V 2x + 1(c) p(x) = x3 .V 2x + 5 1.A 方形的矩陣A 稱相稱如果在= A 和歪曲相稱如果在= .VA 。表示, 如果B 是一個方形的矩陣, 那麼(a) BBT 和B + BT 是symmetric(b) B .V BT 是歪曲相稱。2.Show 如果一個方形的矩陣A 滿足3.A2 .V 2.A + I = 0, 然後A-1 = 2.I .V 3.A 。3.Consider 矩陣A = (a)Find 基本的矩陣E1 和E2 這樣E2.E1A = I. (b)Write A-1 作為二個基本的矩陣(c)Write A 產品作為二個基本的矩陣產品。的4.Prove 如果A 是一個可轉位矩陣並且B 是列等效與A, B 然後是還可轉位的。5.(a) 證明: 如果A 和B 是m .6N n 矩陣, 然後A 和B 是列等值如果和只如果
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