Date | May Example question | Marks available | 2 | Reference code | EXM.1.AHL.TZ0.29 |
Level | Additional Higher Level | Paper | Paper 1 | Time zone | Time zone 0 |
Command term | Show that | Question number | 29 | Adapted from | N/A |
Question
Let M = (ab−ba)(ab−ba) where aa and bb are non-zero real numbers.
Show that M is non-singular.
Calculate M2.
Show that det(M2) is positive.
Markscheme
finding det M =a2+b2=a2+b2 A1
a2+b2>0a2+b2>0, therefore M is non-singular or equivalent statement R1
[2 marks]
M2 = (ab−ba)(ab−ba)=(a2−b22ab−2aba2−b2)(ab−ba)(ab−ba)=(a2−b22ab−2aba2−b2) M1A1
[2 marks]
EITHER
det(M2) =(a2−b2)(a2−b2)+(2ab)(2ab)=(a2−b2)(a2−b2)+(2ab)(2ab) A1
det(M2) =(a2−b2)2+(2ab)2=(a2−b2)2+(2ab)2 (=(a2+b2)2)(=(a2+b2)2)
since the first term is non-negative and the second is positive R1
therefore det(M2) > 0
Note: Do not penalise first term stated as positive.
OR
det(M2) = (det M)2 A1
since det M is positive so too is det (M2) R1
[2 marks]
Examiners report
Syllabus sections
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22M.2.AHL.TZ1.7a.i:
By considering the image of (0, 0)(0, 0), find q.
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22M.2.AHL.TZ1.7c.ii:
Hence find the angle and direction of the rotation represented by R.
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22M.2.AHL.TZ1.7b:
Write down the matrix S.
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22M.1.AHL.TZ2.15b:
Find the value of det M.
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EXM.1.AHL.TZ0.26b:
Calculate det (BA).
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EXM.1.AHL.TZ0.6a:
Find Q.
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EXM.2.AHL.TZ0.23a:
Find the values of λ for which the matrix (A − λI) is singular.
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EXM.1.AHL.TZ0.7b:
Let 3(−4821)−5(20q−4)=(−2224923).
Find the value of q.
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22M.2.AHL.TZ1.7a.ii:
By considering the image of (1, 0) and (0, 1), show that
P=(√34 14-14 √34).
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22M.2.AHL.TZ1.7d.ii:
Find the value of a and the value of b.
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EXM.2.AHL.TZ0.4:
The matrices A, B, X are given by
A = (31−56), B = (480−3), X = (abcd), where a, b, c, d∈Q.
Given that AX + X = Β, find the exact values of a, b, c and d.
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EXM.2.AHL.TZ0.8d.i:
Write down these three equations as a matrix equation.
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21N.2.AHL.TZ0.4b:
Three drones are initially positioned at the points A, B and C. After performing the movements listed above, the drones are positioned at points A′, B′ and C′ respectively.
Show that the area of triangle ABC is equal to the area of triangle A′B′C′ .
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21N.2.AHL.TZ0.4a.iii:
Find P2.
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21N.2.AHL.TZ0.4c:
Find a single transformation that is equivalent to the three transformations represented by matrix P.
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22M.2.AHL.TZ1.7d.i:
Write down an equation satisfied by (ab).
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22M.2.AHL.TZ1.7c.i:
Use P=RS to find the matrix R.
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21N.2.AHL.TZ0.4a.i:
Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.
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21N.2.AHL.TZ0.4a.ii:
Find a single matrix P that defines a transformation that represents the overall change in position.
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21N.2.AHL.TZ0.4a.iv:
Hence state what the value of P2 indicates for the possible movement of the drone.
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EXM.1.AHL.TZ0.30c:
Find the matrix X, such that AX = C.
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EXM.2.AHL.TZ0.8a:
Write down the value of q.
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EXM.1.AHL.TZ0.27a:
(A + B)2 = A2 + 2AB + B2
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EXM.2.AHL.TZ0.23b.iii:
Use the result from part (b) (ii) to explain why A is non-singular.
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EXM.2.AHL.TZ0.10c:
Hence find A−1B.
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EXM.1.AHL.TZ0.27c:
CA = B C = BA
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EXM.1.AHL.TZ0.34:
The square matrix X is such that X3 = 0. Show that the inverse of the matrix (I – X) is I + X + X2.
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18M.1.AHL.TZ0.F_2a:
Show that A4 = 12A + 5I.
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EXM.1.AHL.TZ0.7a.ii:
Find the value of b.
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19M.2.AHL.TZ0.F_3a:
Given that M3 can be written as a quadratic expression in M in the form aM2 + bM + cI , determine the values of the constants a, b and c.
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19M.2.AHL.TZ0.F_3d:
Find a quadratic expression in M for the inverse matrix M–1.
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EXM.2.AHL.TZ0.6a:
Write down A−1.
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EXM.2.AHL.TZ0.6b:
Find D.
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EXM.2.AHL.TZ0.7c.ii:
Find T4.
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EXM.1.AHL.TZ0.45b.i:
Find the matrix DA.
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EXM.1.AHL.TZ0.8a:
Write down the inverse matrix A−1.
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EXM.2.AHL.TZ0.7d:
Using your results from part (a) (ii), find T100.
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EXM.1.AHL.TZ0.9a.ii:
Write down the value of b.
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EXM.2.AHL.TZ0.5b:
find AB.
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EXM.2.AHL.TZ0.7a.i:
Find S4.
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EXM.1.AHL.TZ0.9a.i:
Write down the value of a.
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EXM.2.AHL.TZ0.5a:
write down A + B.
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EXM.1.AHL.TZ0.9b:
Write these equations as a matrix equation.
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EXM.1.AHL.TZ0.8b.i:
Express X in terms of A−1 and B.
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EXM.2.AHL.TZ0.11b:
Write down M−1.
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EXM.2.AHL.TZ0.11a:
Write down the determinant of M.
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EXM.2.AHL.TZ0.20c.ii:
A–1 = I – A.
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EXM.1.AHL.TZ0.54a:
Find det A.
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EXM.1.AHL.TZ0.27b:
(A – kI)3 = A3 – 3kA2 + 3k2A – k3
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EXM.2.AHL.TZ0.8b:
Show that 27m+9n+3p=18.
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EXM.2.AHL.TZ0.3c:
Using this value of a, find M−1 and hence solve the system of equations:
−x+2y=−3
2x−y=3
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EXM.2.AHL.TZ0.20c.i:
A3 = –I.
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EXM.2.AHL.TZ0.6c:
Find X.
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EXM.2.AHL.TZ0.20a.ii:
Show that γ2=γ−1.
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EXM.1.AHL.TZ0.11a:
Find A + B.
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EXM.2.AHL.TZ0.7b.i:
Find M2.
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EXM.1.AHL.TZ0.16b.ii:
Write down M−1 for this value of a.
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EXM.1.AHL.TZ0.4b.i:
Given that XA + B = C, express X in terms of A–1, B and C.
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EXM.1.AHL.TZ0.46:
If A = (x442) and B = (2y84), find 2 values of x and y, given that AB = BA.
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EXM.2.AHL.TZ0.20a.iii:
Hence find the value of (1−γ)6.
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EXM.2.AHL.TZ0.3a:
Find M2 in terms of a.
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EXM.1.AHL.TZ0.45b.ii:
Find B if AB = C.
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EXM.2.AHL.TZ0.22c:
If all of the elements of M are positive, find the range of possible values for a.
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19M.2.AHL.TZ0.F_3b:
Show that M4 = 19M2 + 40M + 30I.
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EXM.1.AHL.TZ0.9c:
Solve the matrix equation.
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EXM.1.AHL.TZ0.16a:
Find a relationship between a and b if the matrices M=(1a23) and N=(1b23) commute under matrix multiplication.
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19M.1.AHL.TZ0.F_13b:
It is now given that A is singular.
By considering appropriate determinants, prove that f is not a bijection.
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EXM.1.AHL.TZ0.3:
A and B are 2 × 2 matrices, where A=[5220] and BA=[112448]. Find B
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EXM.1.AHL.TZ0.29b:
Calculate M2.
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EXM.1.AHL.TZ0.7a.i:
Write down the value of a.
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19M.1.AHL.TZ0.F_13a:
Given that A is non-singular, prove that f is a bijection.
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EXM.2.AHL.TZ0.21a.ii:
does X–1 + Y–1 have an inverse? Justify your conclusion.
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EXM.2.AHL.TZ0.9b:
Hence or otherwise solve
x−3y=1
2x+z=2
4x+y+3z=−1
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EXM.2.AHL.TZ0.8e:
The function f can also be written f(x)=x(x−1)(rx−s) where r and s are integers. Find r and s.
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EXM.1.AHL.TZ0.16b.i:
Find the value of a if the determinant of matrix M is −1.
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EXM.2.AHL.TZ0.21a.i:
find X and Y.
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EXM.2.AHL.TZ0.7b.ii:
Show that M3 = (1601).
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EXM.1.AHL.TZ0.31:
Find the determinant of A, where A = (312958746).
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EXM.2.AHL.TZ0.7c.i:
Write down M4.
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EXM.1.AHL.TZ0.35b:
Hence, find the point of intersection of the three planes.
x−3y+z=12x+2y−z=2x−5y+3z=3
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EXM.2.AHL.TZ0.21b.ii:
and hence find an expression for An+(An)−1.
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EXM.1.AHL.TZ0.28:
Consider the matrix A =(exe−x2+ex1), where x∈R.
Find the value of x for which A is singular.
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EXM.2.AHL.TZ0.11c:
Hence solve M(xy)=(48).
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EXM.2.AHL.TZ0.9a:
Write down the inverse of the matrix A = (1−30201413).
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EXM.1.AHL.TZ0.8b.ii:
Hence, solve for X.
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EXM.1.AHL.TZ0.29a:
Show that M is non-singular.
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EXM.2.AHL.TZ0.10b:
Given that 2A + B = (2643), find the value of p and of q.
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EXM.2.AHL.TZ0.23c:
Use the values from part (b) (i) to express A4 in the form pA+ qI where p, q∈Z.
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EXM.1.AHL.TZ0.54b:
Find the set of values of k for which the system has a unique solution.
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EXM.1.AHL.TZ0.45a:
Given matrices A, B, C for which AB = C and det A ≠ 0, express B in terms of A and C.
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EXM.1.AHL.TZ0.45c:
Find the coordinates of the point of intersection of the planes x+2y+3z=5, 2x−y+2z=7, 3x−3y+2z=10.
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EXM.2.AHL.TZ0.8c:
Write down the other two linear equations in m, n and p.
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EXM.1.AHL.TZ0.30b:
Hence, or otherwise, find A–1.
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19M.2.AHL.TZ0.F_3c:
Using mathematical induction, prove that Mn can be written as a quadratic expression in M for all positive integers n ≥ 3.
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EXM.1.AHL.TZ0.43a:
Find the values of a and b given that the matrix A=(a−4−6−857−534) is the inverse of the matrix B=(12−23b1−11−3).
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20N.1.AHL.TZ0.F_4b.ii:
Assuming that A is non-singular, use the result in part (b)(i) to show that
A-1=1β(αI-A).
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EXM.2.AHL.TZ0.23b.i:
Find the value of m and of n.
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EXM.1.AHL.TZ0.4b.ii:
Given that B =(675−2), and C =(−50−87), find X.
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EXM.2.AHL.TZ0.10a.i:
Find A−1.
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EXM.1.AHL.TZ0.22b:
The matrix C = (2028) and 2AB = C. Find the value of x.
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EXM.1.AHL.TZ0.43b:
For the values of a and b found in part (a), solve the system of linear equations
x+2y−2z=53x+by+z=0−x+y−3z=a−1.
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EXM.1.AHL.TZ0.6c:
Find D–1.
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EXM.2.AHL.TZ0.7a.ii:
Find S100.
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EXM.1.AHL.TZ0.35a:
Write down the inverse of the matrix
A = (1−3122−11−53)
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EXM.1.AHL.TZ0.10b:
det (2A − B).
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EXM.1.AHL.TZ0.6b:
Find CD.
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EXM.1.AHL.TZ0.49:
The square matrix X is such that X3 = 0. Show that the inverse of the matrix (I – X) is I + X + X2.
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EXM.1.AHL.TZ0.11c:
Find AB.
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EXM.2.AHL.TZ0.21b.i:
find x and y in terms of n.
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EXM.1.AHL.TZ0.10a:
2A − B.
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EXM.1.AHL.TZ0.4a:
Write down the inverse, A–1.
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20N.1.AHL.TZ0.F_4b.i:
Verify that
A2-αA+βI= 0.
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EXM.1.AHL.TZ0.44:
Find the values of the real number k for which the determinant of the matrix (k−43−2k+1) is equal to zero.
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EXM.1.AHL.TZ0.52b:
Hence solve the system of equations
x+2y+z=0
x+y+2z=7
2x+y+z=17
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EXM.2.AHL.TZ0.22b:
Hence show that M is a singular matrix.
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EXM.2.AHL.TZ0.8d.ii:
Solve this matrix equation.
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EXM.1.AHL.TZ0.22a:
Find AB.
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EXM.1.AHL.TZ0.35c:
A fourth plane with equation x+y+z=d passes through the point of intersection. Find the value of d.
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EXM.1.AHL.TZ0.52a:
Find the inverse of the matrix (121112214).
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EXM.1.AHL.TZ0.30a:
Given that AB = (a000a000a), find a.
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EXM.1.AHL.TZ0.25a:
Find the matrix A2.
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EXM.2.AHL.TZ0.3b:
If M2 is equal to (5−4−45), find the value of a.
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EXM.1.AHL.TZ0.50:
Given that A = (231−2) and B = (200−3), find X if BX = A – AB.
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EXM.2.AHL.TZ0.22d:
Show that (I − M)2 = I − M where I is the identity matrix.
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EXM.2.AHL.TZ0.23b.ii:
Hence show that I = 15A (6I – A).
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EXM.1.AHL.TZ0.33:
Given that M = (2−1−34) and that M2 – 6M + kI = 0 find k.
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EXM.2.AHL.TZ0.22a.i:
Show that a+d=1.
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EXM.2.AHL.TZ0.22a.ii:
Find an expression for bc in terms of a.
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EXM.1.AHL.TZ0.48:
The matrices A, B, C and X are all non-singular 3 × 3 matrices.
Given that A–1XB = C, express X in terms of the other matrices.
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EXM.1.AHL.TZ0.11b:
Find −3A.
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EXM.2.AHL.TZ0.10a.ii:
Find A2.
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EXM.2.AHL.TZ0.10d:
Let X be a 2 × 2 matrix such that AX = B. Find X.
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20N.1.AHL.TZ0.F_4a:
By considering the determinant of a relevant matrix, show that the eigenvalues, λ, of A satisfy the equation
λ2-αλ+β=0,
where α and β are functions of a, b, c, d to be determined.
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18M.1.AHL.TZ0.F_2b:
Let B = [421−3].
Given that B2 – B – 4I = [k00k], find the value of k.
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EXM.1.AHL.TZ0.2:
If A=(2p3−4pp) and det A=14, find the possible values of p.
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EXM.1.AHL.TZ0.32:
If A = (12k−1) and A2 is a matrix whose entries are all 0, find k.
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EXM.1.AHL.TZ0.26c:
Find A(A–1B + 2A–1)A.
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EXM.1.AHL.TZ0.26a:
Find BA.
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EXM.1.AHL.TZ0.25b:
If det A2 = 16, determine the possible values of a.