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## Numbers and Algebra

### Advanced Algebra - Matrices and Matrix Operations

#### Q.01

'Find the general term of the sequence \ \\left\\{a_{n}\\right\\} \ determined by the following conditions'

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'The sequence {a_{n}} is defined by a_{1}=3, a_{n+1}=2a_{n}-n^{2}+n. Determine the quadratic function f(n) for the sequence {a_{n}-f(n)} to form a geometric progression with a common ratio of 2, and express a_{n} in terms of n.'

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'Exercise 40: Solve the recurrence formulas (1) a_{n+1} = 2a_{n} + b_{n} and (2) b_{n+1} = a_{n} + 2b_{n}. Given initial conditions a_{1} + b_{1} = 4 and a_{1} - b_{1} = 2, find a_{n} and b_{n}.'

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'The inequality holds when 4\\left(a^{2}+b^{2}+c^{2}\\right)\\left(x^{2}+y^{2}+z^{2}\\right) \\geqq(a x+b y+c z)^{2} with equality if \ a y=b x,\\quad b z=c y ,\\quad c x=a z\'

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'Example 37 Recurrence Relation between Adjacent 3 Terms (1)\nFind the general term of the sequence \ \\left\\{a_{n}\\right\\} \ determined by the following conditions.\n(1) \ a_{1}=0, a_{2}=1, a_{n+2}=a_{n+1}+6 a_{n} \\n(2) \ a_{1}=1, \\quad a_{2}=4, \\quad a_{n+2}+a_{n+1}-2 a_{n}=0 \'

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'For a geometric sequence with a positive common ratio, the sum of the first 3 terms is 21, and the sum of the next 6 terms is 1512. Find the first term and the sum of the first 5 terms of this sequence.'

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'If the sum from the first term to the nth term of a sequence {an} is represented by Sn = 3n(n+5), find the general term an.'

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'When m=6, x=-2; when m=10, x=-8,0; when m=-6, x=4; when m=-10, x=2,10'

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'Find the first term a and the common difference d of an arithmetic sequence where the sum of the first 5 terms is 20 and the sum of the first 20 terms is 140.'

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'Find the general term for the sequence {an} determined by the following conditions: a1=-1, an+1=an+4n-1'

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'If the sum of the first n terms of a sequence {a_n} satisfies 3 S_n = a_n + 2n - 1, answer the following questions:'

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'Find the first term and the common ratio of a geometric sequence where the sum of the first three terms is 6 and the sum of the second to fourth terms is -12.'

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'Find the components of two vectors. For the two vectors \\( \\vec{a}=(2,1) \\) and \\( \\vec{b}=(4,-3) \\), determine the components of vectors \ \\vec{x} \ and \ \\vec{y} \ that satisfy the conditions \ \\vec{x}+2\\vec{y}=\\vec{a} \ and \ 2\\vec{x}-\\vec{y}=\\vec{b} \.'

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'In the tetrahedron PABC, let H be the foot of the perpendicular from point A to the plane PBC, and let PA=a, PB=b, PC=c.'

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'61 (1) \ c=-2 \,\n\ x=-1 \ has a minimum value of -5\n(2) \ c=-9 \,\n\ x=-2 \ has a minimum value of -41'

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'In a party with 5 participants, where each person prepares a gift and then draws lots to share them, what is the number of ways in which only two specific individuals, A and B, receive their prepared gifts, while the remaining three receive gifts other than the ones they prepared? Also, the number of ways in which only one person receives their prepared gift is □.'

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'Define the sequence $a_{n}$ as follows. $a_1=1, n a_{n+1}-(n+2) a_{n}+1=0 (n=1,2,3, \\cdots \\cdots)$. Also, define the sequence $b_{n}$ as $b_{n}=2 a_{n}-1 (n=1,2,3, \\cdots \\cdots)$. Furthermore, define the sequence $c_{n}$ as $c_{n}=\x0crac{b_{n}}{n(n+1)} (n=1,2,3, \\cdots \\cdots)$. (1) Express $b_{n+1}$ in terms of $b_{n}$ and $n$. (2) Find the general term of $c_{n}$. (3) Find the general term of $a_{n}$. [Rikkyo University]'

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'Find the general term of the sequence {an} determined by the following conditions.'

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'Given two sequences {a_n} and {b_n} defined as follows, answer the following questions. a_1=4, b_1=1, a_{n+1}=3a_n+b_n, (1), b_{n+1}=a_n+3b_n. (1) Find the general term for sequences {a_n+b_n} and {a_n-b_n}. (2) Find the general term for sequences {a_n} and {b_n}.'

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"Assuming that the profit per 1kg of products P and Q is a million yen and 3 million yen, respectively. Now, let's consider the profit per day. Here, a is assumed to be a positive number. (i) When a = 1, the x, y that maximize profit are (x, y) = (ノハ, ヒフ). (ii) When a takes what value, producing only product Q without producing product P can maximize profit, and the maximum profit at that time is 厼 million yen. (iii) The sufficient and necessary condition for x, y to maximize profit to be only (x, y) = (□テト, ナニ) is ム."

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'(1) Find the general term of the sequence \ \\left\\{a_{n+1}-\\alpha a_{n}\\right\\} \ and the general term of the sequence \ \\left\\{a_{n+1}-\eta a_{n}\\right\\} \, where \ \\alpha \\neq \\ beta \ [Refer to Important Example 41].'

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'Derive three simultaneous difference equations: $a_{n+1}=p a_{n}+q b_{n}, b_{n+1}=r a_{n}+s b_{n}$ (refer to important example 43), obtaining equations of the form $a_{n+1}+\\alpha_{1} b_{n+1}=\eta_{1}(a_{n}+\\alpha_{1} b_{n}), a_{n+1}+\\alpha_{2} b_{n+1}=\eta_{2}(a_{n}+\\alpha_{2} b_{n})$, or deriving recurrence relations involving only $a_{n}$ or $b_{n}$ (recurrence relations between adjacent 3 terms).'

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'The sum of the first to nth terms of the sequence {an} is expressed as Sn=3/4 n(n+3)(n=1,2,3,...). (1) Find an. (2) Prove that ∑(k=1)^(n) k ak is a multiple of 3.'

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"Let's consider the following [problem]. Please find the general term of the sequence \\\left\\{a_{n}\\right\\}\ determined by each condition."

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'Solve the following systems of linear equations.'

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'A parabola y = 2x^{2} + ax + b was parallel shifted 2 units along the x-axis and -3 units along the y-axis, and overlapped with the parabola y = 2x^{2}. Find the values of constants a and b.'

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'Taro, a sprinter in the 100m race, decided to focus on question (1) and think about the best stride and pitch to improve his time.'

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'Given \ \\vec{p}=s \\vec{a}+t \\vec{b}+u \\vec{c} \, the following equations are obtained: \ 2 s+u=1 \ (1), \ -s+3 t=3 \ (2), \ s+2 t+u=2 \. Find the values of s, t, u, and express \ \\vec{p} \.'

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'For matrix A=\\left[ \egin{array}{ll}a & b \\\\ c & d \\end{array} \\right], find its inverse matrix.'

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'Among these matrices, which ones are of the same type? Also, which ones are equal?'

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'For matrices A, B, C, D, answer questions (1) to (3).'

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'This is a problem of finding the inverse matrix.'

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'In general, in matrix multiplication, the commutative law does not hold (AB ≠ BA). Therefore, it is not possible to freely manipulate expressions like polynomials. What can be used unconditionally are the associative law (AB)C = A(BC) and the distributive law (A+B)C = AC + BC, C(A+B) = CA + CB, which can be used to manipulate expressions. For matrices A, B where AB=BA (i.e., commutative), calculations can be performed like ordinary expressions.'

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'Sum of infinite series using recurrence relations'

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'Is there an inverse matrix for the given matrices. If yes, find it.'

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'Inverse matrix\nConditions for the existence of an inverse matrix and its components'

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'Consider the sequence {a_{n}} determined by the conditions (i)(ii).'

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'(1) $s(\\vec{c} \\cdot \\vec{a})+t(\\vec{c} \\cdot \\vec{b})=\\vec{c} \\cdot(s \\vec{a}+t \\vec{b})=\\vec{c} \\cdot \\vec{c}=|\\vec{c}|^{2} \\geqq 0$ Therefore, $s(\\vec{c} \\cdot \\vec{a})+t(\\vec{c} \\cdot \\vec{b}) \\geqq 0$ Reference: Equality holds when $\\vec{c}=\\overrightarrow{0}$.'

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'When A = \\left(\egin{array}{lll}1 & 2 & 4 \\\\ 3 & 1 & 2\\end{array}\\right), B = \\left(\egin{array}{rrr}1 & 1 & 0 \\\\ 0 & -1 & 1 \\\\ 1 & 0 & -1\\end{array}\\right), C = \\left(\egin{array}{ll}1 & 3 \\\\ 2 & 1 \\\\ 4 & 2\\end{array}\\right), choose two different matrices for multiplication and calculate the result.'

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'When the matrices A and B satisfy AB=BA, they are said to be commutative.'

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'Check if there is an inverse matrix for the following matrices. If it exists, find it.'

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'List three basic properties of the inverse matrix.'

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'Maximum value is 9/4 when x=y=3√3; Minimum value is -4 when x=81, y=1/3'

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'Let 28n be a positive integer. Let f(n) be the number of lattice points P(x, y, z) in the xyz space that satisfy the following system of inequalities where x, y, z are all integers, as n approaches infinity. Find the limit of lim_{n -> ∞} f(n)/n^3. The system of inequalities is as follows: { x + y - z <= n, x - y - z <= n, -x - y + z <= n }.'

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'For triangle ABC, let the dot products of vectors AB, BC, and CA be denoted as vector AB·vector BC=x, vector BC·vector CA=y, and 320 vector CA·vector AB=z. Express the area of triangle ABC in terms of x, y, and z.'

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'Question 1, Book p. 609\n(1) A: a 2x2 matrix\nB: a 2x3 matrix\nC: a 3x2 matrix\nD: a 3x3 matrix\n(2) The 3rd row vector is (1, -3), the 2nd column vector is \\(\\left(\egin{array}{r}-1 \\\\ 2 \\\\ -3\\end{array}\\right)\\)\n(3) a_{12} = 5, \\quad a_{32} = -3, \\quad a_{33} = 2'

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'In vectors \ \\vec{x}, \\vec{y} \, when \\( \\vec{x}+2 \\vec{y}=(-2,-4), 2 \\vec{x}+\\vec{y}=(5,-2) \\), find \ \\vec{x} \ and \ \\vec{y} \.'

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'Express the hyperbolic function that passes through the points (-2,3) and (1,6), with the line x=-3 as an asymptote, in the form y=(ax+b)/(cx+d).'

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'Question 6: Perform the following matrix calculations.'

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'Mathematics C\n(2) $\\vec{x} + 2\\vec{y} = \\vec{a}$\n(1), $\\vec{x} - 3\\vec{y} = \\vec{b}$\nLet (2) be\nFrom (1) $\\times 3 +$ (2) $\\times 2$ we get $5\\vec{x} = 3\\vec{a} + 2\\vec{b}$\nTherefore\n$\\vec{x} = \\frac{1}{5}(3\\vec{a} + 2\\vec{b}) = \\frac{1}{5}\\{3(1,1) + 2(1,3)\\} = (1, \\frac{9}{5})$\nAlso, from (1)-(2) we have $5\\vec{y} = \\vec{a} - \\vec{b}$\nHence\n$\\vec{y} = \\frac{1}{5}(\\vec{a} - \\vec{b}) = \\frac{1}{5}\\{(1,1) - (1,3)\\} = (0, -\\frac{2}{5})$\nBy solving the system of equations\n$x, y$\n\\left\\{\egin{\overlineray}{l}\nx + 2y = a \\\nx - 3y = b\n\\end{\overlineray}\\right.\n'

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'Express x and y in terms of a and b, satisfying 2x+5y=a, 3x-2y=b.'

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'Question 5: Please solve the following matrix equation.'

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'Find the angle \ \\theta \ between \ \\vec{a} \ and \ \\vec{b} \ when \ \\vec{a}-\\frac{2}{5} \\vec{b} \ and \ \\vec{a}+\\vec{b} \ are orthogonal, and \ \\vec{a} \ and \ \\vec{a}-\\vec{b} \ are orthogonal.'

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'(2) \\( \egin{array}{l}\\\\left(P^{-1} A P\\right)^{n}=\\left(\egin{array}{ll}3 & 0 \\\\ 0 & 5\\end{array}\\right)^{n} \\\\ \\\\ \\text { Therefore, } \\\\ A^{n}=P\\left(\egin{array}{cc}3^{n} & 0 \\\\ 0 & 5^{n}\\end{array}\\right) P^{-1}\\end{array} \\) So \\( \\quad P^{-1} A^{n} P=\\left(\egin{array}{cc}3^{n} & 0 \\\\ 0 & 5^{n}\\end{array}\\right) \\quad \\angle\\left(P^{-1} A P\\right)^{n}=P^{-1} A^{n} P \\)'

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'Using recurrence equation to find the number of cases.'

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'For each of the following cases, determine if there exists a winning strategy for either player A or B.'

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Perform the following calculations.
(1) 3 ec{a}+2 ec{a}
(2) \( 5 ec{b}-2(-6 ec{b}) \)
(3) \( -2(3 ec{a}-2 ec{b})+4(ec{a}-ec{b}) \)
(4) \( rac{1}{2}(ec{a}+2 ec{b})+rac{3}{2}(ec{a}-2 ec{b}) \)
(5) \( rac{2}{3}(2 ec{a}-3 ec{b})+rac{1}{2}(-ec{a}+5 ec{b}) \)

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Given \(ec{a}=(2,3,1), ec{b}=(2,5,0), ec{c}=(3,1,1)\), express \(ec{p}=(5,10,-1)\) in the form s ec{a}+t ec{b}+u ec{c} using appropriate real numbers $s, t, u$.