If A is m * n matrix such that AB & BA both are defined, then B is a matrix of order
n * m
Matrix Multiplication Principles
To find the order of matrix B, we first need to understand the fundamental condition for matrix multiplication. For the product of two matrices, say P and Q (i.e., PQ), to be defined, the number of columns in the first matrix (P) must be equal to the number of rows in the second matrix (Q).
Matrix AB Definition
We are given that matrix A is an \(m \times n\) matrix. This means:
- Matrix A has \(m\) rows.
- Matrix A has \(n\) columns.
Let's assume the order of matrix B is \(p \times q\). This means:
- Matrix B has \(p\) rows.
- Matrix B has \(q\) columns.
For the product AB to be defined, the number of columns in A must be equal to the number of rows in B.
\[ \text{Number of columns in A} = \text{Number of rows in B} \] \[ n = p \]So, from the condition that AB is defined, we know that matrix B must have \(n\) rows. Its order can be provisionally written as \(n \times q\).
Matrix BA Definition
Next, we are also given that the product BA is defined. Now, consider B as the first matrix and A as the second matrix in this product.
- Matrix B (first in BA) has an order of \(n \times q\), meaning it has \(q\) columns.
- Matrix A (second in BA) has an order of \(m \times n\), meaning it has \(m\) rows.
For the product BA to be defined, the number of columns in B must be equal to the number of rows in A.
\[ \text{Number of columns in B} = \text{Number of rows in A} \] \[ q = m \]So, from the condition that BA is defined, we know that matrix B must have \(m\) columns.
Determining Matrix B's Order
By combining both conditions:
- From AB being defined, we found that matrix B must have \(n\) rows.
- From BA being defined, we found that matrix B must have \(m\) columns.
Therefore, the order of matrix B must be \(n \times m\).
| Product | Order of First Matrix | Order of Second Matrix | Condition for Definition |
|---|---|---|---|
| AB | \(A_{m \times n}\) | \(B_{p \times q}\) | \(n = p\) |
| BA | \(B_{p \times q}\) | \(A_{m \times n}\) | \(q = m\) |
Since \(n=p\) and \(q=m\), the order of matrix B (which is \(p \times q\)) becomes \(n \times m\).
Important Questions from Matrix Algebra
Consider the system of equations: x + y = 2 and 2x + 2y = 5. This system has
The standard ordered basis of R 3 is {e 1, e 2, e 3} Let T : R 3 → R 3 be the linear transformation such that T(e 1) = 7e 1- 5e 3, T (e 2) = -2e 2+ 9e 3, T(e 3) = e 1+ e 2+ e 3. The standard matrix of T is:
The system of equations
x + y + z = 6;
x + 4y + 6z = 20;
x + 4y + λz = μ
has NO solution for values of λ and μ given by
What is the transformation matrix M that transforms a square in the xy-plane defined by (1, 1) T, (-1, 1) T, (-1, -1) T and (1, -1) T to a parallelogram whose corresponding vertices are (2, 1) T, (0, 1) T, (-2, -1) T and (0, -1) T?
- If A = \( \left[\begin{array}{cc}0 & 1 \\ −1 & 0\end{array}\right]\) and (aI 2 + bA)2 = A, then
