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 [tex]$A, 1 \times A=A$[/tex].
For any scalar [tex]$r, r l=l$[/tex], where [tex]$l$[/tex] 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 possible for matrices of any size.
Multiplying a matrix by 1 results in the same matrix: 1 A = A .
Geometric figures can be scaled using scalar multiplication.
Scalar multiplication by an integer is equivalent to repeated addition of the matrix.
The true statements are therefore: statements 1, 2, 4, and 5.
Explanation
Analyzing the Statements We need to evaluate the truthfulness of the given statements about scalar multiplication of matrices. Let's analyze each statement.
Statement 1 Statement 1: You can multiply a matrix of any size by a scalar. This is true because scalar multiplication is defined for any matrix, regardless of its dimensions.
Statement 2 Statement 2: For any matrix A , 1 × A = A . This is true because multiplying any matrix by the scalar 1 results in the original matrix.
Statement 3 Statement 3: For any scalar r , r I = I , where I is the identity matrix. This is false unless r = 1 . For example, if r = 2 , then 2 I = I .
Statement 4 Statement 4: You can scale geometric figures using scalar multiplication. This is true because matrices can represent geometric figures, and scalar multiplication can scale them.
Statement 5 Statement 5: Scalar multiplication is a shortcut for repeated addition of the same matrix. This is true when the scalar is an integer. For example, 3 A = A + A + A .
Statement 6 Statement 6: Scalar multiplication is not possible for matrices that are not square. This is false because scalar multiplication is defined for matrices of any size, including non-square matrices.
Conclusion 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.
Examples
Scalar multiplication is used in image processing to adjust the brightness of an image. Each pixel's color value (represented as a matrix) is multiplied by a scalar to either brighten or darken the image. For example, multiplying by a scalar greater than 1 brightens the image, while multiplying by a scalar between 0 and 1 darkens it. This technique is also fundamental in computer graphics for scaling objects and performing various transformations.
The true statements about scalar multiplication of matrices are: you can multiply any matrix by a scalar, multiplying a matrix by 1 results in the same matrix, scalar multiplication helps scale geometric figures, and it serves as repeated addition for integer scalars. Statements regarding the identity matrix and non-square matrix restrictions are false. Therefore, the true statements are 1, 2, 4, and 5.
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