If X is a matrix of order 3 × 3, Y is a matrix of order 2 × 3 and Z is a matrix of order 3 × 2, then which of the following are correct? 1. (ZY)X is a square matrix having 9 entries. 2. Y(XZ) is a square matrix having 4 entries. 3. X(YZ) is not defined. Select the correct answer using the code given below :
1, 2 and 3
Understanding Matrix Multiplication and Order
This problem involves determining the validity and order of matrix products based on the given dimensions of matrices X, Y, and Z. Recall the rules for matrix multiplication:
- Two matrices A and B can be multiplied (AB) 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 \(m \times n\) and matrix B has an order of \(n \times p\), then the resulting product matrix AB will have an order of \(m \times p\).
Let's list the orders of the given matrices:
- Matrix X: Order \(3 \times 3\)
- Matrix Y: Order \(2 \times 3\)
- Matrix Z: Order \(3 \times 2\)
Now, let's examine each statement.
Analyzing Statement 1: (ZY)X
First, consider the product ZY.
- Z has order \(3 \times 2\).
- Y has order \(2 \times 3\).
The number of columns in Z (2) equals the number of rows in Y (2). So, the product ZY is defined.
The order of the resulting matrix ZY will be \(3 \times 3\).
Next, consider the product (ZY)X.
- The matrix (ZY) has order \(3 \times 3\).
- Matrix X has order \(3 \times 3\).
The number of columns in (ZY) (3) equals the number of rows in X (3). So, the product (ZY)X is defined.
The order of the resulting matrix (ZY)X will be \(3 \times 3\).
A \(3 \times 3\) matrix is a square matrix because the number of rows equals the number of columns. The total number of entries in a \(3 \times 3\) matrix is \(3 \times 3 = 9\).
Therefore, statement 1, "(ZY)X is a square matrix having 9 entries," is correct.
Analyzing Statement 2: Y(XZ)
First, consider the product XZ.
- X has order \(3 \times 3\).
- Z has order \(3 \times 2\).
The number of columns in X (3) equals the number of rows in Z (3). So, the product XZ is defined.
The order of the resulting matrix XZ will be \(3 \times 2\).
Next, consider the product Y(XZ).
- Matrix Y has order \(2 \times 3\).
- The matrix (XZ) has order \(3 \times 2\).
The number of columns in Y (3) equals the number of rows in (XZ) (3). So, the product Y(XZ) is defined.
The order of the resulting matrix Y(XZ) will be \(2 \times 2\).
A \(2 \times 2\) matrix is a square matrix because the number of rows equals the number of columns. The total number of entries in a \(2 \times 2\) matrix is \(2 \times 2 = 4\).
Therefore, statement 2, "Y(XZ) is a square matrix having 4 entries," is correct.
Analyzing Statement 3: X(YZ)
First, consider the product YZ.
- Y has order \(2 \times 3\).
- Z has order \(3 \times 2\).
The number of columns in Y (3) equals the number of rows in Z (3). So, the product YZ is defined.
The order of the resulting matrix YZ will be \(2 \times 2\).
Next, consider the product X(YZ).
- Matrix X has order \(3 \times 3\).
- The matrix (YZ) has order \(2 \times 2\).
The number of columns in X (3) is NOT equal to the number of rows in (YZ) (2). Therefore, the product X(YZ) is not defined.
Statement 3 says, "X(YZ) is not defined." This is correct.
Summary of Matrix Product Orders
| Product | Step 1 Product | Order (Step 1) | Step 2 Product | Order (Step 2) | Defined? | Square Matrix? | Number of Entries |
|---|---|---|---|---|---|---|---|
| (ZY)X | ZY | \(3 \times 3\) | (ZY)X | \(3 \times 3\) | Yes | Yes | 9 |
| Y(XZ) | XZ | \(3 \times 2\) | Y(XZ) | \(2 \times 2\) | Yes | Yes | 4 |
| X(YZ) | YZ | \(2 \times 2\) | X(YZ) | \(3 \times 2\) | No | - | - |
Based on our analysis:
- Statement 1 is correct.
- Statement 2 is correct.
- Statement 3 is correct.
All three statements are correct.
Revision Table: Matrix Operations and Properties
| Concept | Description | Key Rule |
|---|---|---|
| Matrix Order | Number of rows \(m\) by number of columns \(n\), written as \(m \times n\). | - |
| Matrix Multiplication (AB) | Possible only if columns in A = rows in B. | If A is \(m \times n\) and B is \(p \times q\), AB is defined iff \(n=p\). |
| Order of Product (AB) | If A is \(m \times n\) and B is \(n \times p\), then AB is \(m \times p\). | Outer dimensions determine product order. |
| Square Matrix | A matrix with an equal number of rows and columns. | Order is \(n \times n\). |
| Number of Entries | Total elements in a matrix. | For an \(m \times n\) matrix, entries = \(m \times n\). |
Additional Information: Properties of Matrix Multiplication
Matrix multiplication has some important properties:
- Associativity: For matrices A, B, and C where the products are defined, \((AB)C = A(BC)\). This property was used when evaluating products like (ZY)X and Y(XZ). We could compute ZY first and then multiply by X, or compute YZ first and then multiply by X (if defined), etc.
- Non-commutativity: In general, \(AB \neq BA\). The order of multiplication matters.
- Distributivity: \(A(B+C) = AB + AC\) and \((A+B)C = AC + BC\), assuming the additions and multiplications are defined.
Understanding matrix orders and the conditions for multiplication is fundamental to solving problems involving matrix expressions like the one discussed here.
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