Reed Sensor matrix for chess board query
Almost all switch matrix keyboards use normally open / momentary contact switches. Consider this when discussing switch matrix behavior. Your initial (text) description is of a switch matrix only comprising of switches.  For such a matrix, only one switch may be closed at any given moment. This is to prevent aliasing.  A conditions where multiple simultaneous switch closures cause the incorrect key to be detected. And, in some cases, prevents harmful shorts.

Adding a forward biased diode to every switch in a matrix prevents harmful shorts.  For a design where one row is driven high and all other rows are driven low, closing two switches in a single column can directly connect the row driven high with a row driven low in a switch matrix only comprising of switches.  However, if a normally forward biased diode is placed in series with both of these closed switches, only the diode connected to the row driven high would be forward biased.  The diode connected to the row driven low would be reversed biased preventing the harmful short.

Using diodes on every switch is costly.  Most computer keyboard do not use this approach.  Instead they use clever circuitry and layouts to prevent shorts and improve their perceived performance.  In reality, almost all computer keyboard have an n-key-roll-over limit.  A point at which the keyboard fails to detect or distinguish a new pressed key while other keys are already pressed.

For a graphic example of the behavior of a forward and reversed biased diode consider this image from allaboutcircuits.com where the diode on the left is in forward biased and the diode on the right is in reversed bias: