Which of the following statements are true about scalar multiplication of matrices?
You can multiply a matrix of any size by a scalar.
For any matrix $A, 1 \times A=A$.
For any scalar $r, r l=l$, where $l$ is the identity matrix.
You can scale geometric figures using scalar multiplication.
Scalar multiplication is a shortcut for repeated addition of the same matrix.
Scalar multiplication is not possible for matrices that are not square.
Answer (2)
Scalar multiplication is defined for matrices of any size.
Multiplying a matrix by the scalar 1 results in the same matrix.
Scalar multiplication can be used to scale geometric figures represented by matrices.
Scalar multiplication by an integer n is equivalent to adding the matrix to itself n times. Therefore, the true statements are: You can multiply a matrix of any size by a scalar; For any matrix A , 1 × A = A ; You can scale geometric figures using scalar multiplication; Scalar multiplication is a shortcut for repeated addition of the same matrix.
Explanation
Analyzing the Statements We will analyze each statement individually to determine its truth value regarding scalar multiplication of matrices.
Statement 1 Statement 1: You can multiply a matrix of any size by a scalar. This statement is true. Scalar multiplication is defined for matrices of any size.
Statement 2 Statement 2: For any matrix A , 1 × A = A . This statement is true. Multiplying a matrix by the scalar 1 results in the same matrix.
Statement 3 Statement 3: For any scalar r , r l = l , where l is the identity matrix. This statement is false. r I = I is only true when r = 1 . Otherwise, it's false. For example, if r = 2 , then 2 I is not equal to I .
Statement 4 Statement 4: You can scale geometric figures using scalar multiplication. This statement is true. Scalar multiplication can be used to scale geometric figures represented by matrices.
Statement 5 Statement 5: Scalar multiplication is a shortcut for repeated addition of the same matrix. This statement is true. Scalar multiplication by an integer n is equivalent to adding the matrix to itself n times.
Statement 6 Statement 6: Scalar multiplication is not possible for matrices that are not square. This statement is false. Scalar multiplication is possible for any matrix, regardless of whether it is square or not.
Conclusion The true statements are: You can multiply a matrix of any size by a scalar; For any matrix A , 1 × A = A ; You can scale geometric figures using scalar multiplication; Scalar multiplication is a shortcut for repeated addition of the same matrix.
Examples
Scalar multiplication is used in various fields such as computer graphics, physics, and engineering. For example, in computer graphics, scalar multiplication is used to scale objects, change their size, or adjust their brightness. In physics, it's used to multiply vectors by a scalar to change their magnitude, such as scaling a force vector. In engineering, it can be used to adjust the gain of a signal in signal processing.
The true statements about scalar multiplication of matrices are that you can multiply a matrix of any size by a scalar, multiplying by 1 returns the same matrix, you can scale geometric figures, and it acts as repeated addition. The false statements include that it is not limited to square matrices and the specific behavior with the identity matrix. Each property is important in understanding how scalar multiplication operates in linear algebra.
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