The matrix A has x rows and x + 5 columns. The matrix B has y rows and 11 - y columns. Both AB and BA exist. What are the values of x and y respectively?
3 and 8
Understanding Matrix Dimensions for Multiplication
This problem asks us to find the dimensions of two matrices, A and B, given that both matrix products AB and BA exist. The existence of matrix products depends directly on the dimensions of the matrices being multiplied.
Matrix Dimensions and Multiplication Rules
Let's first define the dimensions of the given matrices:
- Matrix A has x rows and x + 5 columns. Its dimension is \(A_{x \times (x+5)}\).
- Matrix B has y rows and 11 - y columns. Its dimension is \(B_{y \times (11-y)}\).
For two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
Condition for AB to Exist
For the matrix product AB to exist, the number of columns in matrix A must be equal to the number of rows in matrix B.
- Number of columns in A = x + 5
- Number of rows in B = y
So, the condition is:
\(x + 5 = y\) (Equation 1)
Condition for BA to Exist
For the matrix product BA to exist, the number of columns in matrix B must be equal to the number of rows in matrix A.
- Number of columns in B = 11 - y
- Number of rows in A = x
So, the condition is:
\(11 - y = x\) (Equation 2)
Solving for x and y using Matrix Multiplication Conditions
We now have a system of two linear equations with two variables:
- \(x + 5 = y\)
- \(11 - y = x\)
We can solve this system using substitution. From Equation 1, we already have y expressed in terms of x: \(y = x + 5\).
Substitute this expression for y into Equation 2:
\(11 - (x + 5) = x\)
Now, simplify and solve for x:
\(11 - x - 5 = x\)
\(6 - x = x\)
Add x to both sides:
\(6 = x + x\)
\(6 = 2x\)
Divide both sides by 2:
\(x = \frac{6}{2}\)
\(x = 3\)
Now that we have the value of x, substitute it back into Equation 1 (or Equation 2) to find y. Using Equation 1:
\(y = x + 5\)
\(y = 3 + 5\)
\(y = 8\)
Verifying the Matrix Dimensions
We found \(x = 3\) and \(y = 8\).
- Matrix A has \(x = 3\) rows and \(x + 5 = 3 + 5 = 8\) columns. So, A is \(3 \times 8\).
- Matrix B has \(y = 8\) rows and \(11 - y = 11 - 8 = 3\) columns. So, B is \(8 \times 3\).
Let's check the matrix multiplication conditions with these dimensions:
- For AB: A is \(3 \times 8\), B is \(8 \times 3\). The number of columns in A (8) equals the number of rows in B (8). AB exists and will be a \(3 \times 3\) matrix.
- For BA: B is \(8 \times 3\), A is \(3 \times 8\). The number of columns in B (3) equals the number of rows in A (3). BA exists and will be an \(8 \times 8\) matrix.
Since both AB and BA exist with \(x=3\) and \(y=8\), these are the correct values.
Conclusion on Matrix Dimensions
The values of x and y that satisfy the conditions for both matrix products AB and BA to exist are \(x=3\) and \(y=8\).
Revision Table: Key Matrix Concepts
| Concept | Explanation | Condition for Multiplication |
|---|---|---|
| Matrix Dimension | Represented as rows \(\times\) columns (e.g., \(m \times n\)). | Not directly involved in the condition, but defines the number of rows and columns. |
| Matrix Multiplication AB | Product of matrix A and matrix B. | Number of columns in A must equal the number of rows in B. If A is \(m \times n\) and B is \(p \times q\), then \(n = p\). The resulting matrix AB will be \(m \times q\). |
| Matrix Multiplication BA | Product of matrix B and matrix A. | Number of columns in B must equal the number of rows in A. If B is \(p \times q\) and A is \(m \times n\), then \(q = m\). The resulting matrix BA will be \(p \times n\). |
Additional Information: Solving Simultaneous Equations for Matrix Problems
Problems involving matrix dimensions often lead to systems of linear equations. Solving these systems accurately is crucial.
- Substitution Method: Solve one equation for one variable, then substitute that expression into the other equation. This reduces the system to a single equation with one variable.
- Elimination Method: Multiply equations by constants so that when the equations are added or subtracted, one variable is eliminated, leaving a single equation with one variable.
- In this problem, the substitution method was straightforward because Equation 1 already gave us an expression for y in terms of x.
Understanding how matrix dimensions affect operations like multiplication is fundamental in linear algebra.
Similar Questions
Consider the following statements in respect of square matrices A, B, C each of same order n :
1. AB = AC ⇒ B = C if A is non-singular
2. If BX = CX for every column matrix X having n rows then B = C
Which of the statements given above is/are correct ?
If \(A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{array}\right]\), then which of the following statements are correct?
1. An will always be singular for any positive integer n.
2. An will always be a diagonal matrix for any positive integer n.
3. An will always be a symmetric matrix for any positive integer n.
Select the correct answer using the code given below:
If \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\), then what is 23A3 - 19A2 - 4A equal to ?
If \(A=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]\), then what is the value of det(I + AA'), where I is the 3 × 3 identity matrix?
Let A and B be non-singular matrices of the same order such that AB = A and BA = B. Which of the following statements is/are correct ?
1. A 2 = A
2. AB 2 = A 2B
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?
Consider the following statements in respect of square matrices A and B of same order :
1. If AB is a null matrix, then at least one of A and B is a null matrix.
2. If AB is an identity matrix, then BA = AB.
Which of the above statements is/are correct?
If A = \(\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & \sin \theta & −\cos \theta\end{array}\right)\) , then which of the following are correct?
1. A + adj A is a null matrix
2. A −1 + adj A is a null matrix
3. A − A −1 is a null matrix
Select the correct answer using the code given below :
\(\rm A=\begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix}\) where a ∈ ℕ, then is A 100 - A 50 - 2A 25 equal to?
where I is the identity matrix.
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?
Important Questions from Operations on Matrices
If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix of the same order, then $AB - BA$ is?
If A and B are two matrices such that AB = B and BA = A, then A 2 + B 2 is equal to
If $A$ is an involuntary matrix and $I$ is a unit matrix of the same order, then $(I + A)^2 - (I - A)^2$ is
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:
Find the additive inverse of matrix A = \(\left[\begin{array}{cc}2 & 1 \\ −3 & 0\end{array}\right]\)
