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COEP Tech University, Pune Department of Mathematics Unit1-Review of matrices Tutorial 1
Part-I: Matrices
(1) Find (a) A + B, (b) 2A − B, (c) AT^ (d) A−^1.
A =
[
]
, B =
[
]
(2) Let A, B, C, D and E be matrices with the provided orders given as A : 3 × 4 , B : 3 × 4 , C : 4 × 2 , D : 4 × 2 , E : 4 × 3. For the following matrices, if defined, determine the size of the matrix; if not defined, provide an explanation.
(a) A + B (b) C + E (c) 12 D (d) − 4 A (e) A − 2 E (f) AC (g) BE (h) E − 2 A (i) 2D + C (j) BET
(3) Verify AB = BA for the following matrices.
(a) A =
[
cos α − sin α sin α cos α
]
, B =
[
cos β − sin β sin β cos β
]
(b) A =
[
]
, B =
[
]
Whether matrix multiplication is commutative in general? Justify.
(4) Find AB and BA, if they are defined.
(a) A =
[
]
, B =
(^) (b) A =
, B =
[
]
(5) Show that a square matrix can be written as sum of a symmetric matrix and a skew- symmetric matrix.
(6) Find inverse of the matrix (if exists).
(a)
[
]
(b)
(7) State whether the following statements are True or False. Justify your answers with proper reason/ counter example. (a) Matrix addition is commutative and associative. (b) Matrix multiplication is neither commutative nor associative. (c) If the matrices A, B, and C satisfy AB = AC, then B = C. (d) (AB)T^ = AT^ BT^. (e) (A + B)T^ = AT^ + BT^. (f) Inverse of non-singular matrix is unique. (g) Product of two invertible matrices is invertible. (h) Sum of two singular matrices is singular. (i) (A−^1 )T^ = (AT^ )−^1. 1
(8) Let A =
[
a b c d
]
. Find products AS and SA for the matrix S =
[
1 x 0 1
]
. Describe in words the effect on A of this product.
(9) Let A be a square matrix. (a) If A^2 = O show that I − A is invertible. (b) If A^3 = O, show that I − A is invertible. (c) In general, if An^ = O for some positive integer n, show that I − A is invertible. [Hint: Think of the geometric series.] (d) Suppose that A^2 + 2A + I = O. Show that A is invertible.
Part-II: Determinants
(1) Find determinant of the matrix.
(a)
[
]
(b)
(^) (c)
(2) Solve for x.
(a)
x + 3 2 1 x + 2
∣ = 0^ (b)
x 0 1 0 x 3 2 2 x − 2
(3) Find |A|, |B|, AB, |AB|, A + B, |A + B|. Verify that (a) |AB| = |A| |B| and (b) |A + B| 6 = |A| + |B|. A =
[
]
, B =
[
]
(4) Let A and B be square matrices of order 4 such that |A| = −5 and |B| = 3. Find (a) |AB| (b) |A^3 | (c) | 3 B| (d) |(AB)T^ | and (e) |A−^1 |.
(5) Use Cramer’s rule to solve the system of equations, if possible. 4 x 1 − x 2 − x 3 = 1, 2 x 1 + 2x 2 + 3x 3 = 10, 5 x 1 − 2 x 2 − 2 x 3 = − 1.
(6) Attempt the following using properties of the determinants.
(a) Find area of triangle with vertices (0, 0), (2, 0), (0, 3).
(b) Determine whether the points (1, 2), (3, 4), (5, 6) are collinear.
(c) Find equation of line passing through the points (− 4 , 7), (2, 4).
(d) Find equation of the plane passing through the points (1, − 2 , 1), (− 1 , − 1 , 7), (2, − 1 , 3).
(e) Find volume of the tetrahedron having vertices (1, 1 , 1), (0, 0 , 0), (2, 1 , −1), (− 1 , 1 , 2).
(f) Determine whether (1, 2 , 3), (− 1 , 0 , 1), (0, − 2 , −5), (2, 6 , 11) are coplanar.
