If A is a matrix of order 3 × 5 and B is a matrix of order 5 × 3, then the order of AB and BA will respectively be
3 × 3 and 5 × 5
Understanding Matrix Multiplication Order
To determine the order of the product of two matrices, say AB, we first need to check if the multiplication is possible. Matrix multiplication AB is defined only if the number of columns in matrix A is equal to the number of rows in matrix B.
If matrix A has an order of \(m \times n\) (meaning \(m\) rows and \(n\) columns) and matrix B has an order of \(p \times q\) (meaning \(p\) rows and \(q\) columns), the product AB is defined if \(n = p\).
If the multiplication is defined (i.e., \(n = p\)), the resulting matrix AB will have an order of \(m \times q\). The number of rows in AB is the number of rows in A, and the number of columns in AB is the number of columns in B.
Calculating the Order of AB
We are given:
- Matrix A has an order of \(3 \times 5\). Here, \(m=3\) and \(n=5\).
- Matrix B has an order of \(5 \times 3\). Here, \(p=5\) and \(q=3\).
For the product AB to be defined, the number of columns in A must equal the number of rows in B.
Number of columns in A is 5.
Number of rows in B is 5.
Since \(5 = 5\), the multiplication AB is defined.
The order of the resulting matrix AB is given by (number of rows in A) \(\times\) (number of columns in B).
Order of AB = \(m \times q = 3 \times 3\).
Calculating the Order of BA
Now let's determine the order of the product BA. For the product BA to be defined, the number of columns in matrix B must equal the number of rows in matrix A.
We have:
- Matrix B has an order of \(5 \times 3\). Here, the number of columns is 3.
- Matrix A has an order of \(3 \times 5\). Here, the number of rows is 3.
For the product BA to be defined, the number of columns in B must equal the number of rows in A.
Number of columns in B is 3.
Number of rows in A is 3.
Since \(3 = 3\), the multiplication BA is defined.
The order of the resulting matrix BA is given by (number of rows in B) \(\times\) (number of columns in A).
Order of BA = \(p \times n = 5 \times 5\).
Summary of Matrix Product Orders
Based on our calculations:
- The order of matrix AB is \(3 \times 3\).
- The order of matrix BA is \(5 \times 5\).
Therefore, the order of AB and BA will respectively be \(3 \times 3\) and \(5 \times 5\).
| Product | Order Calculation | Resulting Order |
|---|---|---|
| AB | \((\text{Rows of A}) \times (\text{Columns of B}) = 3 \times 3\) (since columns of A = rows of B = 5) | \(3 \times 3\) |
| BA | \((\text{Rows of B}) \times (\text{Columns of A}) = 5 \times 5\) (since columns of B = rows of A = 3) | \(5 \times 5\) |
Revision Table: Matrix Order Basics
| Concept | Description |
|---|---|
| Matrix Order | Represented as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. |
| Condition for AB | Number of columns in A must equal the number of rows in B. |
| Order of AB | If A is \(m \times n\) and B is \(n \times q\), then AB is \(m \times q\). |
| Condition for BA | Number of columns in B must equal the number of rows in A. |
| Order of BA | If B is \(p \times q\) and A is \(q \times n\), then BA is \(p \times n\). |
Additional Information: Matrix Multiplication Properties
Matrix multiplication is a fundamental operation in linear algebra. Understanding the order of the resulting matrix is crucial. Here are a few additional points about matrix multiplication:
- Non-commutative: In general, AB \(\neq\) BA. As seen in this problem, even if both AB and BA are defined, their orders might be different, or if the orders are the same, the resulting matrices typically have different elements.
- Associative: Matrix multiplication is associative, meaning \((AB)C = A(BC)\), provided the multiplications are defined.
- Distributive: Matrix multiplication is distributive over matrix addition, meaning \(A(B+C) = AB + AC\) and \((A+B)C = AC + BC\), provided the operations are defined.
- Identity Matrix: There exists an identity matrix, denoted by I, such that for any matrix A, \(AI = IA = A\), provided the multiplication is defined. The identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere. Its order depends on the dimensions of A to make the multiplication conformable.
The order of a matrix product is determined solely by the dimensions of the original matrices, not their specific elements.
Similar Questions
Consider the following in respect of matrices A, B and C of same order:
1) (A + B + C)' = A' + B’ + C’
2) (AB)’ = A’B’
3) (ABC)’ = C’B’A’
Where A’ is the transpose of the matrix A.
Which of the above are correct?If \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 1&0&{ - 2}\\ 2&{ - 3}&4 \end{array}} \right]\) , then the matrix X for which 2X + 3A = 0 holds true is
Let \(A = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}\) and (mI + nA) 2= A where m, n are positive real numbers and I is the identify matrix. What is (m + n) equal to?
Let \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {\rm x + y}& \rm y\\ {\rm 2x}&{\rm x - y} \end{array}} \right],\;\rm B = \left[ {\begin{array}{*{20}{c}} 2\\ { - 1} \end{array}} \right]\) and \(\rm C = \left[ {\begin{array}{*{20}{c}} 3\\ 2 \end{array}} \right]\) . If AB = C, then what is the value of the determinant of the matrix A?
If \(A = \left[ {\begin{array}{c} 1&{ - 1}\\ { - 1}&1 \end{array}} \right],\) then the expression A 3- 2A 2is
If \(A = \left( {\begin{array}{*{20}{c}} 1&2\\ 2&3\\ 3&4 \end{array}} \right)\) and \(B = \left( {\begin{array}{*{20}{c}} 1&2\\ 2&1 \end{array}} \right),\) then which one of the following is correct?
A square matrix A is called orthogonal if_______ where A’ is the transpose of A.
Consider the following in respect of matrices A and B of same order:
1) A 2– B 2= (A + B) (A – B)
2) (A – I) (I + A) = O ⇔ A 2= I
Where I is the identity matrix and O is the null matrix.
Which of the above is/are correct?If \({\rm{f}}\left( {\rm{x}} \right) = \left[ {\begin{array}{c} {\cos {\rm{x}}}&{ - \sin {\rm{x}}}&0\\ {\sin {\rm{x}}}&{\cos {\rm{x}}}&0\\ 0&0&1 \end{array}} \right]\) , then which of the following are correct?
1. f(θ) × f(ϕ) = f(θ + ϕ)
2. The value of the determinant of the matrix f(θ) × f(ϕ) is 1
3. The determinant of f(x) is an even function.
Select the correct answer using the code given below:
Consider the following in respect of the matrices:
A = [m n], B = [-n -m] & \( C = \begin{bmatrix} m \\ -m\end{bmatrix} \)
1. CA = CB
2. AC = BC
3. C(A + B) = CA + CB
Which of the above statements is/are correct?
Important Questions from Operations on Matrices
The adjoint of matrix \(\left[ {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right]\)is
Consider the following in respect of matrices A, B and C of same order:
1) (A + B + C)' = A' + B’ + C’
2) (AB)’ = A’B’
3) (ABC)’ = C’B’A’
Where A’ is the transpose of the matrix A.
Which of the above are correct?If \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 1&0&{ - 2}\\ 2&{ - 3}&4 \end{array}} \right]\) , then the matrix X for which 2X + 3A = 0 holds true is
Find a matrix X such that 2A + B + X = 0 , where
\(A=\begin{bmatrix} -1 & 2 \\\ 3 & 4 \end{bmatrix} \ \text{and} \;\rm B =\ \begin{bmatrix} 3 & -2 \\\ 1 & 5 \end{bmatrix} \ ?\)
The solution of the matrix equation \(\left[ {\begin{array}{*{20}{c}} 2&{ - 1}&3\\ 1&1&1\\ 1&{ - 1}&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x\\ y\\ z \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 9\\ 6\\ 2 \end{array}} \right]\) is:
