Date | May Example question | Marks available | 2 | Reference code | EXM.1.AHL.TZ0.27 |
Level | Additional Higher Level | Paper | Paper 1 | Time zone | Time zone 0 |
Command term | Write down | Question number | 27 | Adapted from | N/A |
Question
Let A, B and C be non-singular 2×2 matrices, I the 2×2 identity matrix and k a scalar. The following statements are incorrect. For each statement, write down the correct version of the right hand side.
(A + B)2 = A2 + 2AB + B2
(A – kI)3 = A3 – 3kA2 + 3k2A – k3
CA = B C = BA
Markscheme
(A + B)2 = A2 + AB + BA + B2 A2
Note: Award A1 in parts (a) to (c) if error is correctly identified, but not corrected.
[2 marks]
(A – kI)3 = A3 – 3kA2 + 3k2A – k3I A2
Note: Award A1 in parts (a) to (c) if error is correctly identified, but not corrected.
[2 marks]
CA = B ⇒ C = BA–1 A2
Note: Award A1 in parts (a) to (c) if error is correctly identified, but not corrected.
[2 marks]
Examiners report
Syllabus sections
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22M.2.AHL.TZ1.7a.i:
By considering the image of (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 .
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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 =
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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 .
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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 .
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EXM.2.AHL.TZ0.5b:
find AB.
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EXM.2.AHL.TZ0.7a.i:
Find 4.
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EXM.1.AHL.TZ0.9a.i:
Write down the value of .
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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 .
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EXM.2.AHL.TZ0.3c:
Using this value of , find and hence solve the system of equations:
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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 .
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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 for this value of .
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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 = and B = , find 2 values of and , given that AB = BA.
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EXM.2.AHL.TZ0.20a.iii:
Hence find the value of .
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EXM.2.AHL.TZ0.3a:
Find in terms of .
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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 .
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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 and if the matrices and 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:
and are 2 × 2 matrices, where and . Find
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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 .
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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
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EXM.2.AHL.TZ0.8e:
The function can also be written where and are integers. Find and .
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EXM.1.AHL.TZ0.16b.i:
Find the value of if the determinant of matrix 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 = .
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EXM.1.AHL.TZ0.31:
Find the determinant of A, where A = .
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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.
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EXM.2.AHL.TZ0.21b.ii:
and hence find an expression for .
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EXM.1.AHL.TZ0.28:
Consider the matrix A , where .
Find the value of for which A is singular.
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EXM.2.AHL.TZ0.11c:
Hence solve M.
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EXM.2.AHL.TZ0.9a:
Write down the inverse of the matrix A = .
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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 = , find the value of and of .
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EXM.2.AHL.TZ0.23c:
Use the values from part (b) (i) to express A4 in the form A+ I where , .
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EXM.1.AHL.TZ0.54b:
Find the set of values of 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 , , .
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EXM.2.AHL.TZ0.8c:
Write down the other two linear equations in , and .
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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 and given that the matrix is the inverse of the matrix .
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20N.1.AHL.TZ0.F_4b.ii:
Assuming that is non-singular, use the result in part (b)(i) to show that
.
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EXM.2.AHL.TZ0.23b.i:
Find the value of and of .
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EXM.1.AHL.TZ0.4b.ii:
Given that B , and C , 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 = and 2AB = C. Find the value of .
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EXM.1.AHL.TZ0.43b:
For the values of and found in part (a), solve the system of linear equations
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EXM.1.AHL.TZ0.6c:
Find D–1.
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EXM.2.AHL.TZ0.7a.ii:
Find 100.
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EXM.1.AHL.TZ0.35a:
Write down the inverse of the matrix
A =
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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 and in terms of .
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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
0.
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EXM.1.AHL.TZ0.44:
Find the values of the real number for which the determinant of the matrix is equal to zero.
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EXM.1.AHL.TZ0.52b:
Hence solve the system of equations
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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 passes through the point of intersection. Find the value of .
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EXM.1.AHL.TZ0.52a:
Find the inverse of the matrix .
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EXM.1.AHL.TZ0.30a:
Given that AB = , find .
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EXM.1.AHL.TZ0.25a:
Find the matrix A2.
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EXM.2.AHL.TZ0.3b:
If is equal to , find the value of .
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EXM.1.AHL.TZ0.50:
Given that A = and B = , 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 = A (6I – A).
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EXM.1.AHL.TZ0.33:
Given that M = and that M2 – 6M + kI = 0 find k.
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EXM.2.AHL.TZ0.22a.i:
Show that .
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EXM.2.AHL.TZ0.22a.ii:
Find an expression for in terms of .
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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, , of satisfy the equation
,
where and are functions of to be determined.
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18M.1.AHL.TZ0.F_2b:
Let B = .
Given that B2 – B – 4I = , find the value of .
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EXM.1.AHL.TZ0.2:
If and det , find the possible values of .
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EXM.1.AHL.TZ0.29c:
Show that det(M2) is positive.
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EXM.1.AHL.TZ0.32:
If A = and A2 is a matrix whose entries are all 0, find .
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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 .