Show that AB = BA where, A = [-23-1-121-69-4],B=[13-122-130-1] - Mathematics and Statistics

Question

Show that AB = BA where,

A = $[\begin{matrix} - 2 & 3 & - 1 \\ - 1 & 2 & - 1 \\ - 6 & 9 & - 4 \end{matrix}], B = [\begin{matrix} 1 & 3 & - 1 \\ 2 & 2 & - 1 \\ 3 & 0 & - 1 \end{matrix}]$

Sum

Solution

AB = $[\begin{matrix} - 2 & 3 & - 1 \\ - 1 & 2 & - 1 \\ - 6 & 9 & - 4 \end{matrix}] [\begin{matrix} 1 & 3 & - 1 \\ 2 & 2 & - 1 \\ 3 & 0 & - 1 \end{matrix}]$

= $[\begin{matrix} - 2 + 6 - 3 & - 6 + 6 - 0 & 2 - 3 + 1 \\ - 1 + 4 - 3 & - 3 + 4 - 0 & 1 - 2 + 1 \\ - 6 + 18 - 12 & - 18 + 18 + 0 & 6 - 9 + 4 \end{matrix}]$

∴ AB = $[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ ...(1)

BA = $[\begin{matrix} 1 & 3 & - 1 \\ 2 & 2 & - 1 \\ 3 & 0 & - 1 \end{matrix}] [\begin{matrix} - 2 & 3 & - 1 \\ - 1 & 2 & - 1 \\ - 6 & 9 & - 4 \end{matrix}]$

= $[\begin{matrix} - 2 - 3 + 6 & 3 + 6 - 9 & - 1 - 3 + 4 \\ - 4 - 2 + 6 & 6 + 4 - 9 & - 2 - 2 + 4 \\ - 6 + 0 + 6 & 9 + 0 - 9 & - 3 + 0 + 4 \end{matrix}]$

∴ BA = $[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ ...(2)

From (1) and (2), we get,

AB = BA.

shaalaa.com

Matrices - Properties of Matrix Multiplication

Is there an error in this question or solution?

Q 4. (i)Q 3 Q 4. (ii)

Chapter 4: Determinants and Matrices - Exercise 4.6 [Page 94]

APPEARS IN

Balbharati Mathematics and Statistics 1 (Arts and Science) [English] 11 Standard Maharashtra State Board

Chapter 4 Determinants and Matrices
Exercise 4.6 | Q 4. (i) | Page 94

RELATED QUESTIONS

If A = $[\begin{matrix} 1 & - 3 \\ 4 & 2 \end{matrix}], B = [\begin{matrix} 4 & 1 \\ 3 & - 2 \end{matrix}]$ show that AB ≠ BA.

Show that AB = BA where,

A = $[\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}], B = [\begin{matrix} \cos ϕ & - \sin ϕ \\ \sin ϕ & \cos ϕ \end{matrix}]$

If A = $[\begin{matrix} 4 & 8 \\ - 2 & - 4 \end{matrix}]$ , prove that A² = 0

Verify A(BC) = (AB)C of the following case:

A = $[\begin{matrix} 2 & 4 & 3 \\ - 1 & 3 & 2 \end{matrix}], B = [\begin{matrix} 2 & - 2 \\ 3 & 3 \\ - 1 & 1 \end{matrix}], C = [\begin{matrix} 3 & 1 \\ 1 & 3 \end{matrix}]$

Verify that A(B + C) = AB + AC of the following matrix:

A = $[\begin{matrix} 1 & - 1 & 3 \\ 2 & 3 & 2 \end{matrix}], B = [\begin{matrix} 1 & 0 \\ - 2 & 3 \\ 4 & 3 \end{matrix}], C = [\begin{matrix} 1 & 2 \\ - 2 & 0 \\ 4 & - 3 \end{matrix}]$

If A = $[\begin{matrix} 1 & - 2 \\ 5 & 6 \end{matrix}], B = [\begin{matrix} 3 & - 1 \\ 3 & 7 \end{matrix}]$ , Find AB - 2I, where I is unit matrix of order 2.

If A = $[\begin{matrix} 4 & 3 & 2 \\ - 1 & 2 & 0 \end{matrix}], B = [\begin{matrix} 1 & 2 \\ - 1 & 0 \\ 1 & - 2 \end{matrix}]$ show that matrix AB is non singular

A = $[\begin{matrix} α & 0 \\ 1 & 1 \end{matrix}], B = [\begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}]$ find α, if A² = B.

If A = $[\begin{matrix} 3 & 4 \\ - 4 & 3 \end{matrix}] and B = [\begin{matrix} 2 & 1 \\ - 1 & 2 \end{matrix}]$ , show that (A + B)(A – B) = A² – B²

If A = $[\begin{matrix} 1 & 2 \\ - 1 & - 2 \end{matrix}], B = [\begin{matrix} 2 & a \\ - 1 & b \end{matrix}]$ and if (A + B)² = A² + B² . find values of a and b

Find matrix X such that AX = B, where A = $[\begin{matrix} 1 & - 2 \\ - 2 & 1 \end{matrix}]$ and B = $[\begin{matrix} - 3 \\ - 1 \end{matrix}]$

Find x, if $[\begin{matrix} 1 & x & 1 \end{matrix}] [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5 \end{matrix}] [\begin{matrix} 1 \\ - 2 \\ 3 \end{matrix}]$ = 0

Find x, y, z if ${3 [\begin{matrix} 2 & 0 \\ 0 & 2 \\ 2 & 2 \end{matrix}] - 4 [\begin{matrix} 1 & 1 \\ - 1 & 2 \\ 3 & 1 \end{matrix}]} [\begin{matrix} 1 \\ 2 \end{matrix}]$ = $[\begin{matrix} x - 3 \\ y - 1 \\ 2 z \end{matrix}]$

If A = $[\begin{matrix} \cos α & \sin α \\ - \sin α & \cos α \end{matrix}]$ , show that $A^{2} = [\begin{matrix} \cos 2 α & \sin 2 α \\ - \sin 2 α & \cos 2 α \end{matrix}]$

If A = $[\begin{matrix} 1 & 2 \\ 3 & 5 \end{matrix}]$ B = $[\begin{matrix} 0 & 4 \\ 2 & - 1 \end{matrix}]$ , show that AB ≠ BA, but |AB| = IAl·IBI

Jay and Ram are two friends in a class. Jay wanted to buy 4 pens and 8 notebooks, Ram wanted to buy 5 pens and 12 notebooks. Both of them went to a shop. The price of a pen and a notebook which they have selected was Rs.6 and Rs.10. Using Matrix multiplication, find the amount required from each one of them

Select the correct option from the given alternatives:

If A + B = $[\begin{matrix} 7 & 4 \\ 8 & 9 \end{matrix}]$ and A − B = $[\begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix}]$ then the value of A is _______

Select the correct option from the given alternatives:

If $[\begin{matrix} x & 3 x - y \\ zx + z & 3 y - w \end{matrix}] = [\begin{matrix} 3 & 2 \\ 4 & 7 \end{matrix}]$ then ______