Question

Matrix $X$ has $r$ rows and $c$ columns, and matrix $Y$ has $c$ rows and $d$ columns, where $r, c,$ and $d$ are different. Which of the following statements must be false? I. The product $Y X$ exists II. The product of $X Y$ exists and has $r$ rows and $d$ columns. III. The product $X Y$ exists and has $c$ rows and $c$ columns. (A) I only (B) II only (C) III only (D) I and II (E) I and III

Name: Problem 28, Chapter :
Uploaded: 2019-08-31T15:30:15-08:00
Duration: 1 min 31 s
Channel: Dilip Paruchuri
Description: Matrix X has r rows and c columns, and matrix Y has c rows and d columns, where r, c, and d are different. Which of the following statements must be false? I. The product Y X exists II. The product of X Y exists and has r rows and d columns. III. The product X Y exists and has c rows and c columns. (A) I only (B) II only (C) III only (D) I and II (E) I and III

   Matrix $X$ has $r$ rows and $c$ columns, and matrix $Y$ has $c$ rows and $d$ columns, where $r, c,$ and $d$ are different. Which of the following statements must be false?
I. The product $Y X$ exists
II. The product of $X Y$ exists and has $r$ rows and $d$ columns.
III. The product $X Y$ exists and has $c$ rows and $c$ columns.
(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III

Instant Answer

Solved by Expert Dilip Paruchuri

08/31/2019

Step 1

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. Show more…

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Matrix $X$ has $r$ rows and $c$ columns, and matrix $Y$ has $c$ rows and $d$ columns, where $r, c,$ and $d$ are different. Which of the following statements must be false? I. The product $Y X$ exists II. The product of $X Y$ exists and has $r$ rows and $d$ columns. III. The product $X Y$ exists and has $c$ rows and $c$ columns. (A) I only (B) II only (C) III only (D) I and II (E) I and III

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Key Concepts

Matrix Product Dimensions

When multiplying two matrices that meet the compatibility requirement, the dimensions of the resulting product are given by the number of rows of the first matrix and the number of columns of the second matrix. This key idea underlies the determination of whether a specified product has the stated dimensions.

Matrix Multiplication Compatibility

Matrix multiplication is defined when the number of columns in the first matrix equals the number of rows in the second matrix. This condition determines whether the product is even possible and influences the assessment of statements regarding the existence of such matrix products.

Order of Matrix Multiplication

Matrix multiplication is generally not commutative, which means that even if the product AB exists, the product BA may not exist, or may have different dimensions. This concept is essential when considering products like XY versus YX.

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