Home
Class 12
MATHS
if A=[{:(4,0,-3),(1,2,0):}]and B=[{:(2,1...

`if A=[{:(4,0,-3),(1,2,0):}]and B=[{:(2,1),(1,-2),(3,4):}]'`then find AB and BA.

Text Solution

AI Generated Solution

The correct Answer is:
To find the products \( AB \) and \( BA \) for the given matrices \( A \) and \( B \), we will follow the matrix multiplication rules step by step. ### Given Matrices: \[ A = \begin{pmatrix} 4 & 0 & -3 \\ 1 & 2 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 1 \\ 1 & -2 \\ 3 & 4 \end{pmatrix} \] ### Step 1: Find \( AB \) 1. **Determine the dimensions of the matrices**: - \( A \) is a \( 2 \times 3 \) matrix. - \( B \) is a \( 3 \times 2 \) matrix. - The product \( AB \) will be a \( 2 \times 2 \) matrix. 2. **Calculate the elements of \( AB \)**: - The element at position (1,1): \[ AB_{11} = 4 \cdot 2 + 0 \cdot 1 + (-3) \cdot 3 = 8 + 0 - 9 = -1 \] - The element at position (1,2): \[ AB_{12} = 4 \cdot 1 + 0 \cdot (-2) + (-3) \cdot 4 = 4 + 0 - 12 = -8 \] - The element at position (2,1): \[ AB_{21} = 1 \cdot 2 + 2 \cdot 1 + 0 \cdot 3 = 2 + 2 + 0 = 4 \] - The element at position (2,2): \[ AB_{22} = 1 \cdot 1 + 2 \cdot (-2) + 0 \cdot 4 = 1 - 4 + 0 = -3 \] 3. **Combine the results**: \[ AB = \begin{pmatrix} -1 & -8 \\ 4 & -3 \end{pmatrix} \] ### Step 2: Find \( BA \) 1. **Determine the dimensions of the matrices**: - \( B \) is a \( 3 \times 2 \) matrix. - \( A \) is a \( 2 \times 3 \) matrix. - The product \( BA \) will be a \( 3 \times 3 \) matrix. 2. **Calculate the elements of \( BA \)**: - The element at position (1,1): \[ BA_{11} = 2 \cdot 4 + 1 \cdot 1 = 8 + 1 = 9 \] - The element at position (1,2): \[ BA_{12} = 2 \cdot 0 + 1 \cdot 2 = 0 + 2 = 2 \] - The element at position (1,3): \[ BA_{13} = 2 \cdot (-3) + 1 \cdot 0 = -6 + 0 = -6 \] - The element at position (2,1): \[ BA_{21} = 1 \cdot 4 + (-2) \cdot 1 = 4 - 2 = 2 \] - The element at position (2,2): \[ BA_{22} = 1 \cdot 0 + (-2) \cdot 2 = 0 - 4 = -4 \] - The element at position (2,3): \[ BA_{23} = 1 \cdot (-3) + (-2) \cdot 0 = -3 + 0 = -3 \] - The element at position (3,1): \[ BA_{31} = 3 \cdot 4 + 4 \cdot 1 = 12 + 4 = 16 \] - The element at position (3,2): \[ BA_{32} = 3 \cdot 0 + 4 \cdot 2 = 0 + 8 = 8 \] - The element at position (3,3): \[ BA_{33} = 3 \cdot (-3) + 4 \cdot 0 = -9 + 0 = -9 \] 3. **Combine the results**: \[ BA = \begin{pmatrix} 9 & 2 & -6 \\ 2 & -4 & -3 \\ 16 & 8 & -9 \end{pmatrix} \] ### Final Results: \[ AB = \begin{pmatrix} -1 & -8 \\ 4 & -3 \end{pmatrix}, \quad BA = \begin{pmatrix} 9 & 2 & -6 \\ 2 & -4 & -3 \\ 16 & 8 & -9 \end{pmatrix} \]
To find the products \( AB \) and \( BA \) for the given matrices \( A \) and \( B \), we will follow the matrix multiplication rules step by step. ### Given Matrices: \[ A = \begin{pmatrix} 4 & 0 & -3 \\ 1 & 2 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} ...
Promotional Banner

Topper's Solved these Questions

  • MATRICES

    NAGEEN PRAKASHAN ENGLISH|Exercise Exercise 3c|10 Videos

  • MATRICES

    NAGEEN PRAKASHAN ENGLISH|Exercise Exercise 3d|3 Videos

  • MATRICES

    NAGEEN PRAKASHAN ENGLISH|Exercise Exercise 3a|20 Videos

  • LINEAR PROGRAMMING

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|9 Videos

  • PROBABIILITY

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|19 Videos

Similar Questions

Explore conceptually related problems

if A=[{:(1,2,3),(4,5,6):}]and B=[{:(-3,-2),(0,1),(-4,-5):}], then find AB and BA ,

if A=[{:(4,2),(-3,2),(1,3):}]and B=[{:(-1,3),(0,2),(2,-4):}], then find 3A-4B.

If A=[{:(2,1,2),(1,2,4):}] and B=[{:(4,1),(2,3),(1,2):}] Find AB and BA ( if it exist)

If A=[{:(2,3,-1),(1,4,2):}] and B=[{:(2,3),(4,5),(2,1):}] then AB and BA are defined and equal.

if A=[{:(3,2,1),(1,2,3),(3,-6,1):}],B=[{:(1,4,0),(2,-3,0),(1,2,0):}] and C=[{:(1,2,3),(3,2,1),(8,7,9):}], then find AB-AC.

If A=[{:(,-2,3),(,4,1):}] and B=[{:(,1,2),(,3,5):}] , find (i) AB (ii) BA (iii) Is AB=BA? (iv) Write the conclusion that you draw from the result obtained above in (iii).

if A=[{:(3,-1,2),(0,5,-3),(1,-2,7):}]and B=[{:(1,0,0),(0,1,0),(0,0,1):}], find whether AB=BA or Not .

if 2A -3B =[{:(4,2),(-1,0),(3,-2):}]and 3A+B=[{:(1,0),(3,5),(-1,4):}] , then find the matrices A And B,

If A=[{:(0,-1,2),(4,3,-4):}] and B=[{:(4,0),(1,3),(2,6):}] then verify that (i) (A')'=A (ii) (AB)'=B'A'

If A=[{:(,1),(,2),(,3):}]and B =[{:(,-5,4,0),(,0,2,-1),(,1,-3,2):}]"then AB"